•NRLF
MONOGRAPHS ON PHYSICS
EDITED BY
SIR J. J. THOMSON, O.M., F.R.S.
MASTER OF TRINITY COLLEGE, CAMBRIDGE
AND
FRANK HORTON, Sc.D.
PROFESSOR OF PHYSICS IN THE UNIVERSITY OF LONDON
MONOGRAPHS ON PHYSICS.
Edited by SIR J. J. THOMSON, O.M., F.R.S.,
Master of Trinity College, Cambridge,
and FRANK HORTON, Sc.D.,
Professor of Physics in the University of London.
THE SPECTROSCOPY OF THE EXTREME ULTRA-
VIOLET. By THEODORE LYMAN, Ph.D., Assistant Professor
of Physics in Harvard University. With Diagrams.
RELATIVITY, THE ELECTRON THEORY, AND GRAVI-
TATION. By E. CUNNINGHAM, M.A., Fellow and Lecturer
of St. John's College, Cambridge.
THE EMISSION OF ELECTRICITY FROM HOT BODIES.
By O. W. RICHARDSON, F.R.S., Wheatstone Professor of
Physics, King's College, London.
MODERN SEISMOLOGY. By G. W. WALKER, A.R.C.Sc.,
F.R.S., Deputy University Lecturer in Astrophysics in the
University of Cambridge. With Plates and Diagrams.
RAYS OF POSITIVE ELECTRICITY AND THEIR APPLI-
CATION TO CHEMICAL ANALYSIS. By SIR J. J.
THOMSON, O.M., F.R.S., Master of Trinity College, Cam-
bridge. With Illustrations.
LONGMANS, GREEN AND CO.
39 PATERNOSTER ROW, LONDON
NEW YORK, BOMBAY, CALCUTTA, AND MADRAS
RELATIVITY
THE ELECTRON THEORY
AND
GRAVITATION
BY
E. CUNNINGHAM, M.A.
FELLOW AND LECTURER OF ST. JOHN'S COLLEGE, CAMBRIDGE
WITH DIAGRAMS
SECOND EDITION
LONGMANS, GREEN AND CO,
39 PATERNOSTER ROW, LONDON
FOURTH AVENUE & 30-TH STREET, NEW YORK
BOMBAY, CALCUTTA, AND MADRAS
1921
PRINTED IN GREAT BRITAIN BY THE UNIVERSITY PRESS, ABERDEEN
PREFACE TO SECOND EDITION.
THE first edition of this book was published while the
General Principle of Relativity was being worked out, before
it seemed possible to arrive at any confirmation from obser-
vation. Shortly after, however, it was shown that the new
theory explained the motion of the perihelion of Mercury, and
now the result of the Solar Eclipse expedition has clinched
matters.
It seemed best to leave practically untouched the account
of the special principle as the natural avenue of approach to
the broader view, which is described at length in Part II.
Acknowledgment is due to the author's friends, Prof. H. F.
Baker and Mr. E. V. Appleton, who kindly read through
the proofs of the new matter.
E. C
CAMBRIDGE, 1920.
491358
PREFACE TO FIRST EDITION.
THIS monograph is an attempt to set out as clearly and simply
as possible the relation of the Principle of Relativity to the
generally accepted Electron Theory, showing at what points
the former is the natural and necessary complement of the
latter.
No attempt has been made to describe to any great extent
consequences of the Principle which would be for the most
part beyond the reach of experimental investigation, and the
mathematical analysis has been omitted as far as possible with
the hope of rendering the account useful to the general reader,
especially to the experimental physicist. Those who desire to
follow out the train of thought in more detail, and especially
to make acquaintance with the mathematical presentation
developed by Minkowski, may be referred to the author's
larger book on the same subject. The author's acknow-
ledgments are due to the Cambridge University Press for
their permission to use some of the material of that work
in the preparation of this.
E. C.
CAMBRIDGE, 1915.
vi
CONTENTS.
PAGE
Preface to Second Edition ... v
Preface to First Edition . vi
Mote on Vector Notation i
CHAPTER
I. Introductory « 2
PART I.
THE SPECIAL PRINCIPLE.
II. The Origin of the Principle 8
III. The Relativity of Space and Time 28
IV. The Relativity of the Electro-Magnetic Vectors ... .46
V. Mechanics and the Principle of Relativity . . . .61
VI. Minkowski's Four-Dimension Vectors 72
PART II.
THE GENERAL PRINCIPLE OF RELATIVITY.
VII. The General Theory 86
VIII. Verification of Einstein's Theory 113
IX. Further Generalization, Weyl's Theory of Electricity . . 125
Index 147
NOTE ON VECTOR NOTATION.
VECTOR quantities are denoted throughout the book by
Clarendon Type, thus, e, h, u.
The components of vectors are denoted by italic type, thus
e has components (ex, ey) ez\ F has components (Fx, Fy, Fz).
Definitions.
(i) Vector Product. — The vector product of two vectors,
e, h, is a vector at right angles to the directions of both, and
of magnitude equal to the product of the magnitudes of the
two into the sine of the angle between them. It is denoted
by the symbol [eh].
The components of [eh] are
(eyhs - ejiy,ejtx - ejtztejty - ejix).
(ii) Divergence of a vector.
~be~ ~tiev ^er
div e = — - + ~~y + —
<>*• ty tz
= limit SS*«flS/V as V -» o
where </S is an element of a closed surface bounding a volume
V, and en is the component of e normal to */S.
(iii) Curl of a vector.
curl c • (^L — v ^L ^ **y — A ^
V " ^' ^ " ^' ^ " ty*
The component of curl e in a given direction is
equal to limit $£5<&/S as S -> o
where ds is an element of the arc of a closed curve which
bounds a small area S to which the given direction is normal.
CHAPTER I.
INTRODUCTORY.
I . THE term Principle of Relativity was first applied to the
hypothesis that it is impossible by means of physical experi-
ments to determine the absolute velocity of a body through
space. This was introduced by Einstein in 1905 not as a
metaphysical doctrine based on the supposition that it is
impossible to conceive of an absolute standard of position in
space. Such an impossibility had often been asserted by
writers on the theory of mechanics, but it was quite irrelevant
to Einstein's hypothesis. Physical theory has nothing to do
with an absolute space. The space with which it deals is one
aspect of the relations which have been observed between the
phenomena with which that theory is concerned.
Newton in setting out his scheme of dynamics postulated
an absolute standard of position in space relative to which all
velocities are measured ; such a beginning was suggested by
the history of the astronomical problem with which he was
dealing. Copernicus shifted the centre of the universe from the
earth to the sun, and from this it was a very short step to the
assumption that the sun was no more likely to be an absolute
centre than the earth. Hence the assumption of an unknown,
though still definite, criterion of position in space.
But when, on the background of this assumption, the science
of dynamics had been developed, it was found that the laws
which had been framed had a very special property. If they
were universally satisfied, there was no means of determining
the actual velocity of any body, though the relative velocity of
two bodies was determinate. In fact, it in no way disturbs the
2
I NT ROD UCTOR Y 3
general laws of dynamics if an arbitrary velocity is added to
the velocity of every body in the system. In other words, the
frame of reference assumed at the beginning is not a unique
one, but may be any one of an infinite number, of which any
one has relative to any other a constant velocity of translation
without rotation. To assert that a unique frame of reference
can never be found is perhaps not absolutely proved, but
neither is any scientific hypothesis. All that has been done is
that the hypothesis has been put on such a footing that it
seems a waste of time to endeavour to dispense with it.
This dynamical relativity is a very different thing from a
statement as to what the mind is able to conceive about
position and motion. It is indeed doubtful whether it is
possible or logical to think about either of these entirely apart
from the regularities perceived in the motions of actual bodies.
In any case, if the doctrine of the relativity of all motion were
#z£ta-physical, it would have to extend beyond the scope of
that indicated by Newton's laws of motion, and to deny the
existence of any criterion of a fixed direction in space or of the
validity of the conception of a constant velocity. It would
result in an entirely agnostic position about motion which
would have to be superseded whenever it was desired to deal
with actual facts of experience. We shall see later that
Einstein has in his later work been able to show that the facts,
as they are known to us, are as a matter of fact entirely
consistent with such a position.
2. The limited scope of the relativity arising out of dyna-
mical theory has always been rather unsatisfying to the mind,
and so it was almost with a sense of relief that the rise of the
electro-magnetic theory of light, bringing with it the suggestion
of a universal aether, was hailed as offering a possibility of a
return to an absolute theory, that is, to one in which a velocity
can be uniquely assigned to every body, namely, its velocity
relative to the aether. In fact, it seemed clear, since all
astronomical observations are made by optical means, that the
frame of reference in practice must be actually the medium
relative to which light is propagated. Thus arose the attempt
to determine the velocity of the earth relative to the aether.
4 RELATIVITY AND THE ELECTRON THEORY
Such a quest has of course no meaning apart from a de-
finite conception of the aether. In the first instance the experi-
ments were rather directed towards clearing up the question as
to what was the best way to think of the aether, whether as an-
other variety of matter filling all space not otherwise occupied,
or as a medium of such nature that it could penetrate matter and
be undisturbed by the passage of matter through it. It was
the latter way of thinking of it toward which opinion gradu-
ally tended and which had been almost universally adopted.
It is perhaps not wise to make a definite statement as to why
this was so, but, at any rate, a very important factor was the
development of the precise statement of the laws of electro-
magnetic phenomena as initiated by Maxwell and elaborated
in the electron theory of Larmor and Lorentz.
Just as the enunciation of Newton's laws of motion gave
a meaning to the notion of ' absolute direction ' which before it
did not possess, so the adoption of Maxwell's equations, as ex-
pressing the laws of propagation of light and electro-magnetic
disturbances, gave a new meaning to the term ' position in
space' by relating it to the medium whose properties the
equations were thought to embody. We must remember,
however, that this medium is defined only by these same pro-
perties and equations, and that to base any argument on an
analogy between it and a hypothetical continuous medium of
material and mechanical properties in the ordinary sense —
properties which have not been shown to be involved in the
definition of the aether — is to become liable to finding ourselves
at variance with the facts. The aether is, in truth, nothing
more than the aggregate of the functions which it serves,
though we find it difficult to think of it except in some concrete
and uniquely existent form ; the exact form, however, which
is adopted is largely determined by individual preference for
this or that analogy.
3. The conception of the 'stagnant' aether having been
adopted, it came as a great surprise that all experimental
efforts failed to find a physical effect due to the motion of
bodies through it. Theory seemed to indicate several ways
of observing such an effect. The failure of all these methods
INTRODUCTORY 5
necessitated a reconciliation of theory and experiment, and
such a reconciliation was rendered possible by the work of
Larmor and Lorentz ; if not complete in every detail, yet it
went so far that many were led to the belief that not only
some but all experiments of this nature were foredoomed to
failure (e.g. Larmor, Brit. Ass., Belfast, 1902).
To admit such a possibility is to allow that the aether,
whatever its nature, may for ever remain concealed. This
is to weaken considerably our power of thinking of the aether
as an actually existing medium. But it in no way affects the
scheme of properties which it was intended to connote.
Rather in the hands of Lorentz and Larmor it becomes a con-
firmation of the theory to find that the experiments actually
do fail. But it is hardly true to say that, if we grant the hypo-
thesis that we shall never be able to identify a unique frame of
reference for the mathematical theory, then we deny the exist-
ence of a real medium of propagation of the physical effects of
light and electricity. What we do is only to question the
sufficiency of the existing conceptions of the nature of this
medium, and the validity of identifying it with a frame of
reference which experience finds to be far from definite.
4. If the dynamics of Newton left some ambiguity in the
meaning of space relations there is no such doubt left in the
case of time intervals. To him a time interval was so definite
that he was able to speak of absolute time as ' flowing evenly
on ' as if independent of all phenomena. One of the funda-
mental elements in this idea of absolute time is that of the
' simultaneity of events ' at different places. No ambiguity
about the meaning of this appears to have arisen.
But if we grant the impossibility of determining the
velocity of the earth through the aether, using the term for a
moment in the sense of the frame of reference for the propa-
gation of light, this idea of * simultaneity ' becomes as vague as
that of the velocity of a moving body. If we consider that
the science of astronomy and the law of gravitation are
dependent for their verification on optical observations, we
shall be inclined to agree that so far at any rate the criterion
of simultaneity which has been used in practice has been one
6 RELATIVITY AND THE ELECTRON THEORY
based on optical communication. Now we shall see later that
the setting up of a standard of simultaneity by means of light
signals is not possible until a definite velocity is assigned to
the observer. Thus the hypothesis of relativity requires a
reconsideration of the way in which we measure time.
This again reacts on the measurement of the length of a
material body, the ' distance between two points ' being the
distance between simultaneous positions of those points. Thus
it becomes necessary also to re-examine the way in which we
measure space. It becomes impossible to consider space and
time separately ; the two measures are inter-related to such
an extent that Minkowski felt himself constrained to say that
" from henceforth time by itself and space by itself are mere
shadows, that they are only two aspects of a single and
indivisible manner of co-ordinating the facts of the physical
world".1
5. In what follows, then, we have to consider first the exact
way in which our fundamental notions of space, time, and
motion have to be modified. This will lead us on to a state-
ment of the status assigned to dynamical theory according to
the principle of relativity, and in particular will require a discus-
sion of the ideas of momentum and energy. It is well known
that the idea of a constant mass for every body has to be
modified if we take into account the effects of radiation and
electrical constitution. We shall see that we can draw, from
the general Principle of Relativity, some very important con-
clusions as to the mechanical relations of systems, without
having recourse to any particular theory of the nature of an
electron or of the way in which matter is built up out of them.
Our growing sense of the insufficiency of fhe existing pictures
of the constitution of matter makes this a very important
consideration.
The following out in detail of the consequences of Einstein's
work of 1905, should by loosening our hold on the Newtonian
methods of thought, so deeply ingrained, make it easier to take
the further step to the appreciation of the much more general
and satisfying position which underlies Einstein's more recent
1 Raum und Zeit, Leipzig, 1909.
INTRODUCTORY 7
work. To this we shall return later. For the moment, we
can but record the fact, that, setting out to examine the
objection of the philosopher to the extremely limited scope of
his so-called Principle of Relativity, he not only was able to
satisfy the philosopher, but at the same time to explain facts
which had refused to bow to Newtonian dynamics, and to lead
to the entirely new discovery that rays of light are bent by
gravitation. To this we shall come in Part II.
PART I.
THE SPECIAL PRINCIPLE OF RELATIVITY.
CHAPTER II.
THE ORIGIN OF THE PRINCIPLE.
6. SPACE AND TIME IN NEWTON'S DYNAMICS.
IN order to appreciate the full significance of the new point of
view introduced by Einstein in 1905 with the technical name,
" The Principle of Relativity " now more precisely called " The
Special Principle of Relativity," it is desirable to make a
survey of the previous lines of thought.
Newton planted deeply in scientific thought the ideas of
an absolute position in space and of an absolute time. The
significance of the Copernican revolution in astronomy extends
beyond the moving of the origin of reference from the observer
to the sun. It makes it impossible to think even of the sun
as a permanent and ultimate point of reference. We naturally
go on to think of the sun as one member of a system greater
than the solar system, and we can see no end to the process
of shifting the point of reference. We have no conception of
the actual velocity of the earth or of the sun in space. But
the precise scheme of laws set forth by Newton introduced an
absolute element into our conceptions of celestial motions. In
the light of his laws we perceive that the reason that we are
able over a large range of terrestrial phenomena to treat the
earth as the point of reference and to ignore the remainder of the
universe, is just that we may think of the earth's absolute velocity
as changing very slowly. Similarly, the reason that a theory
of the solar system as a separate system is possible, ignoring
8
THE ORIGIN OF THE PRINCIPLE g
the whole of the rest of the universe, is that we may conceive
of its centre of mass as having a constant absolute velocity.
Although later we find that the ' absolute velocity ' of a body
is not determinable, yet the classical dynamics do give us a
criterion of something which we may call 'absolute direction,'
though the philosopher may assert that we are not using the term
in a strict sense. In the sense in which we use the term this
absolute direction is not a thing that is obvious a priori, arising
out of pure thought ; it is a fact which emerges out of the
perceived motions of bodies as we try to reduce them to the
simplest possible descriptive scheme. The scientific concep-
tion of space is inseparable from the laws of physics. There
is no other way of making space an object of measurement
save by means of this body of law. Apart from it or from
some equivalent, motion is a mere sensation, though it is
doubtful whether the most primitive man is without some
rudimentary notion of those regularities which have been
summed up in the law of inertia.
7. THE RELATIVITY OF NEWTON'S DYNAMICS.
But now we come to the remarkable fact that the absolute-
ness introduced into our ideas of space by the laws of dynamics
is only partial. It is a commonplace that if any frame of re-
ference is known with respect to which the laws of motion are
satisfied then any other frame which is moving relative to
that known frame with a constant velocity as judged by the
standards of dynamics, is equally suitable as a basis for the
description of the motion by the same laws.
This partial relativity it will be convenient to speak of as
' Newtonian Relativity '. It is not a philosophic principle, it
is purely empirical, arid rests for its justification on the agree-
ment of the deductions from it with the facts of observation.
Thus from this point of view we have no criterion whereby we
may say in any defined sense that a given point is at rest in space.
A similar position is to be taken as to the nature of time
as an object of measurement. There may or may not be an
a priori standard of the equality of two elements of duration,
but we do not know what it may be. For scientific purposes
io RELATIVITY AND THE ELECTRON THEORY
of exact measurement the time of which we think is defined
by the laws of motion. There is no real need to say with
Newton that " absolute time flows uniformly on," and to
pretend that, given this time, we lay our phenomena of motion
in thought against it and so measure them. We cannot doubt
that the definition of time was invented after the system of
dynamics had been worked out, and was prefixed to satisfy
the demand for definitions and postulates at the beginning of
any systematic treatise.
8. TIME NOT AN ABSOLUTE OR INDEPENDENT CONCEPT.
Time, then, as an object of measurement is nothing more
than one aspect of the relations which we have been able to
disentangle from the complex of phenomena of motion. But
to our everyday life it is such a definite aspect that it seems
to exist almost in its own right. The partial relativity that
exists in the scheme of relations leaves no ambiguity in the
term 'equality of two intervals of time' nor in the term
* equality of two distances '. These are such definite and
separable elements in the construction of the scheme that it is
possible to reconstruct the idea of the measurement of motion
out of them. The Newtonian picture of the motions of bodies
is the perfected result of a long process of analysis of percept-
ions and refinement of concepts. Beginning with the crude
perception of change, the continual comparison of motions
results in the extraction of the conceptual time and space of
mathematics. In terms of these we are able to build up a
mental model of the motion which is susceptible to mathe-
matical treatment. This model is only approximate, since it
necessarily ignores certain factors in the actual system. Our
planetary theory, for instance, disregards the effect of the
distant stars, and the neglect is justified a posteriori by the fact
that the discrepancy between model and system is incapable
of observation.
9. MATHEMATICAL FORM OF THE NEWTONIAN
PRINCIPLE OF RELATIVITY.
Before proceeding further, it will be useful to set down in
mathematical form the chief aspects of 'Newtonian Relativity'.
THE ORIGIN OF THE PRINCIPLE 1 1
(i) The Space-Time Transformations. — Let (x, y, z) be a set
of space co-ordinates and /an associated time variable, defining
together a Newtonian frame of reference — that is, which are
such that if the motion is described by the use of these
variables, the laws of motion are satisfied Let a new set of
variables be taken, defined by the equations
x = x - at, y = y - fit, z = z - 7*, f = t.
Now let the motion be described by turning the given relations
between (#, y,z\ and t into relations between (x\ y\ z'} and /,
then the relations so obtained between (xt y ', z) and / will
also satisfy the laws of motion ; in other words, the new
variables also define a Newtonian frame of reference.
A point which as measured in the first set of co-ordinates
has the velocity (u, v, w) has in the second the velocity (u - a,
v — /3, w — 7), and the relative velocity of two points is the
same in both systems, a, /3, 7 being given constants.
(ii) The Law of Addition of Velocities. — The relations
u = u - a, v = v - /?, w = w - 7
connecting the velocities of a given point relative to two New-
tonian frames of reference, of which the origin of the second
has a velocity (a, /Q, 7) relative to the first, seem to us to be
obvious relations ; but it is necessary to remember that the fact
that either (uy v, w) or (u, v, w) may be conveniently called
the velocity of the body in the appropriate frame of reference
depends upon what has been said of the possibility of main-
taining unchanged the form of the dynamical laws when we
change the frame of reference. We shall see later that these
relations cease to be convenient when we are dealing with the
extremely great velocities which we associate with the nega-
tive electrons constituting the cathode and /3-rays.
(iii) Newtonian Relativity consists in the fact that either
(x, y, z, f) or (x, /, z\ /) are equally valid as space-time co-
ordinates, and (u, v, w) or (u, v, w'} as the corresponding
measures of velocity. But there are certain quantities which
are ' invariants/ that is, which have the same value which-
ever set of co-ordinates we employ. These are — to mention the
most simple — the ' mass ' of a particle, the ' force ' acting upon
12 RELATIVITY AND THE ELECTRON THEORY
a particle, and the ' acceleration ' of a particle. When we come
to higher dynamics, we find that the ' constants of inertia ' of a
rigid body and other quantities have invariant values, whereas
the ' energy ' of a system is a relative quantity though the ' prin-
ciple of the conservation of energy ' is an invariant relation.
10. THE PRINCIPLE OF CONSERVATION OF MOMENTUM
DEDUCIBLE FROM THE PRINCIPLES OF CONSERVA-
TION OF ENERGY AND OF RELATIVITY.
If we assume that the principle of the conservation of
energy in the form
' rate of increase of kinetic energy = rate of work of forces '
is true whatever Newtonian frame of reference we use, we may
deduce the principle that
* rate of increase of momentum in any direction — sum of forces
in that direction ' .
For, whatever the values of the constants a, ft, 7, we must
^m{(u - of + (v - /3)2 + (w - 7)2}
= 2{X(« - a) + Y(z, - £) + Z(w - 7)},
m being the mass of a particle of the system, and (X, Y, Z)
being the components of any force applied to the system.
Hence
>. du , dv ,
= 2 {X(« - a) + Y(v - £) + Z(w - 7},
whatever the values of a, /3, 7.
Hence the coefficients of a, /3, 7 separately must be equal
on the two sides of the equation : or
^ du Vv
*m Tt = 2X
^ dw ^^
2m — -- 2Z,
which equations express exactly the ordinary principle of
momentum.
THE ORIGIN OF THE PRINCIPLE 13
Exactly a similar relation between the three principles of
momentum, energy, and relativity is maintained in Einstein's
Special Principle of Relativity (1905) which is to be developed
later.
ii. THE ADVENT OF AETHER THEORY.
It is on the basis of time and space as described above
that subsequent physical thought has rested. All motion has
been tacitly referred to a Newtonian frame of reference. But
the nineteenth century began a new era. Early in that cen-
tury the idea of a luminiferous aether came into prominence.
The question was at once asked, " What is the velocity of the
earth relative to this medium ? "
Arago, sometime before iSiS,1 devised an experiment
which he thought would answer that question. On the wave
theory of light the index of refraction of a piece of glass is the
ratio of the velocity of the incident light to the velocity of the
refracted light. If the glass were moving through the aether
to meet the incident light, the relative velocity of the light at
entering the prism would be greater than if the prism were at
rest in the aether. The velocity of the light in the prism
should, however, be the same, so Arago thought. Thus the
index of refraction and therefore the deviation of the ray of
light, should be greater when the prism is moving towards the
ray than when it is moving away from the ray or is at rest in
the aether. It was such a difference in the deviation that
Arago hoped to find when he examined rays of light from
stars in different directions, all of which could not make the
same angle with the motion of the earth through the aether.
The difference corresponding to a velocity of the earth equal
to its velocity relative to the sun would have amounted to
about one minute of angle, and this was well within the reach
of his means of observation, so that he expected to perceive a
measurable difference ; but he did not. Writing to Fresnel for
an opinion on his experiments, he was told that the only place at
which his argument might break down was in the assumption
1 See the letter from Fresnel to Arago on the subject of the experiments
" Annales de Chimie," 1818.
14 RELATIVITY AND THE ELECTRON THEORY
that the light travelled through the prism at the same relative
rate whether the prism was moving or at rest in the aether.
Fresnel suggested that the prism did not communicate to the
light the whole of its velocity but only a fraction of it. If u
were the velocity of the light in the prism when at rest in the
aether, and the prism were set in motion with velocity v relative
to the aether, then Fresnel asserted that the deviation would
not be altered at all, if the velocity of the light relative to the
prism were now (u - v/jj,2), fj, being the ordinary index of re-
fraction ; or, in other words, if the prism communicated to the
ray not its whole velocity but only the fraction (l - I///,2) of it.
This fraction is now commonly spoken of as * Fresnel's
convection-coefficient \1
12. MATTER MODIFIED BY ITS MOTION THROUGH THE
AETHER.
The point of Fresnel's suggestion which is of most signi-
ficance for us here is that it implies that matter, by its motion
through the aether, may be modified in just such a way as to
neutralize an effect which would otherwise arise. We shall
see later several more instances of hypotheses that have been
suggested for the purpose of explaining the non-appearance of
effects to which theory seemed to point. It was one of the
main objects of the 1905 Principle of Relativity to gather
together these separate ad hoc hypotheses under a single
assumption, and to deduce consequences which do not arise
out of the particular and disconnected hypotheses.
13. FIZEAU'S VERIFICATION OF FRESNEL'S HYPOTHESIS.
In 1851 Fizeau set out to verify directly Fresnel's sugges-
tion of a convection-coefficient by actually observing the effect
of the motion of, a stream of water on the velocity of light
through it. He devised an experiment in which a beam of
light is divided into two parts which traverse two parallel tubes
filled with water which can be set in motion with a measur-
able velocity. A beam of light from the source a (Fig. I)
1 Larmor, " £Sther and Matter," 1900, p. 38, shows that Fresnel's hypo-
thesis is not only sufficient but is necessary if Arago's experimental evidence is
universally confirmed.
THE ORIGIN OF THE PRINCIPLE 15
rendered parallel by the lens b is divided into two parts by the
half-silvered plate c. The two parts are reflected by the
mirrors d into the two tubes AB, CD which contain the water,
and are sent back through the opposite tube by the totally
reflecting prism/; they are then reunited by the mirrors d and
the plate c and enter the telescope g, through which inter-
ference fringes are observed.
d
FIG i. — Fizeau's apparatus as improved by Michelson and Morley.
Explanation of Diagram. — a, source of light ; b, condensing lens; c, half-silvered
plate of glass to divide the beam of light ; d, d, mirrors ; e, e, tubes carrying
stream of water in opposite directions ; /, totally reflecting prism ; g, ob-
serving telescope.
The water in the tubes is then caused to circulate, moving
in opposite directions along AB and CD, so that it is moving
with one part of the beam of light and against the other. A
shifting of the interference fringes is at once observed, which
is proportional to the velocity of the water.
Supposing that the velocity of light through the moving
water is c + kv, where c is the velocity when the velocity v is
zero, the times taken by the two parts of the beam to traverse
the total path / in the water are respectively
the velocity v being in opposite senses for the two parts.
1 6 RELATIVITY AND THE ELECTRON THEORY
Thus the retardation of the one beam relative to the other is
/ / 2lkv
c - kv c ' + kv c'2
neglecting (v/c)*.
On measuring the displacement of the interference fringes,
Fizeau found that Fresnel's value for the convection-coeffici-
ent k, viz. I - fju - 2, fitted his results quite well. The con-
clusion was confirmed later by Michelson and Morley, who re-
peated the experiment with modern refinements. The detailed
discussion of Michelson and Morley's results will be referred to
later (p. 41, §§ 36-8). It will be seen that they are entirely in
agreement with the predictions of the principle of relativity.
14. THE SIGNIFICANCE OF FRESNEL'S CONVECTION-CO-
EFFICIENT, AND OF FlZEAU'S VERIFICATION OF IT.
The remarkable experimental verification of Fresnel's bril-
liant guess considerably strengthened the tendency towards
the "stagnant aether" theory. It might have seemed that the
hypothesis was a very artificial one, but we must admit that
Fresnel's preference for this conception was not without reason.
To quote his own words : —
" If we admitted that our globe communicates its velocity
to the aether which surrounds it, it would be easy to see why
the same prism refracts light in the same manner from what-
ever direction it comes. But it seems impossible to explain
the aberration of the stars on this hypothesis ; so far I have
not been able to think clearly of this phenomenon except by
supposing that the aether passes freely through the earth, and
that the velocity given to this subtle fluid is only a very small
part of that of the earth ; not more than one hundredth part,
for example."
The outstanding fact of aberration must, indeed, not be
forgotten in considering the various experiments and hypo-
theses that have been devised to meet the various questions
that arise.
Sir George Stokes, a great student of the motion of fluids,
tried hard to evolve a theory of the aether which would allow
THE ORIGIN OF THE PRINCIPLE 17
of it being dragged along with the earth, and at the same time
be consistent with the law of aberration. But his theory never
received a large amount of support, and was practically aban-
doned when the establishment of the identity of light and
electric disturbances required that the aether should be con-
ceived as much in relation to electro-magnetic phenomena as
to optical. But the immediate significance of Fizeau's ex-
periment was that it was thereby definitely shown that the
velocity of light relative to a material medium is not a con-
stant depending on the nature of the medium alone, but is
also dependent on the velocity of the medium through space,
or rather through the aether, which was conceived by Fresnel
to be at rest in space.
The effect being a first order one, that is, being propor-
tional to the first power of the velocity, the experiment throws
no light on the velocity of the earth itself, since the part of
the effect due to this velocity remains practically constant, the
observed effect being proportional to the change in the velo-
city of the water in the tubes relative to the earth. But
henceforth it was clear that in respect of its optical properties
it must be admitted that a material medium is in some sense
modified by its motion.
When at a later date light was identified as an electro-
magnetic disturbance this became one deciding factor as be-
tween Hertz' theory of electro-magnetic phenomena in moving
bodies, and that of Maxwell and Lorentz. Hertz' idea of a
moving body was just an extension of the Newtonian idea of
a rigid body in motion. The properties of the body were con-
ceived by him to be entirely unchanged when it was set in
motion, just as the dynamical mass of a body was unaltered ;
and among other implications was the complete convection
of light, that is, the invariance of the velocity of light relative
to the body. In view of Fizeau's experiment it was not pos-
sible to maintain this theory, and other experiments soon
showed it to be lacking in other respects.
Fresnel's suggestion of the interpenetration of aether and
matter therefore assumed greater prominence, and became the
basis of the later theory developed by Lorentz and by Larmor.
2
i8
RELATIVITY AND THE ELECTRON THEORY
15. THE MICHELSON-MORLEY EXPERIMENT.
The question was, however, by no means regarded as
settled by Fizeau's results. In 1881 we find A. A. Michelson
returning to it with an attempt at a direct investigation of the
difference in the velocity of light relative to the earth in dif-
ferent directions by a method which did not involve propaga-
tion through material bodies. It was hoped thus to prevent
the possibility of an explanation of the results by means of
hypotheses about the modification of matter by its motion
through the aether such as Fresnel had advanced in the case
of Arago's experiment. The result of this experiment on
subsequent thought was so far-reaching that it will be well to
describe it in detail.
The experiment was first carried out by Michelson in
1 88 1, and was repeated with greater refinement with the
assistance of E. W. Morley1 in 1887, and again with even
greater care by Morley and D. B. Miller in I9O5.2
Fizeau's experiment, as has been pointed out, being a first
order experiment, would only reveal the influence of velocity
relative to the earth. In the Michelson- Morley experiment
it was proposed to seek for an effect depending on the
square of the velocity, and to
avoid any question concerning
the internal constitution of
moving matter. The arrange-
M2 ment was as follows, the figure
showing the path of the light
relative to the moving ap-
paratus : —
A beam of light from a
source S was divided by
partial reflection at a plate of glass A into two portions
travelling along the paths AMX, AM2. These two portions
were reflected back by mirrors Mj and M2, and, on striking
the plate again, a portion of the beam from M1 is transmitted
and brought to interference with a portion of the beam from
C
FIG. 2.
Phil. Mag.," 1887.
2 Ibid., 1905.
THE ORIGIN OF THE PRINCIPLE 19
M2 reflected along AC. The whole apparatus could be rotated
into any position desired.
1 6. THE THEORY OF THE EXPERIMENT.
Suppose first that AM2 is in the direction of the earth's
velocity relative to the aether, which will be called v. Then
since (c - v) is the velocity of light relative to the apparatus
on the forward journey and (c + v} is the relative velocity on
the return journey, the time taken by the light to travel from
A to M2 and back is
C - V C + V < - V
where /2 is the distance AM2.
In considering the time taken to go to Mj and back we
have to take a ray whose velocity relative to the apparatus is
along AMj, which is assumed perpendicular to AM2. The
relative velocity along this line will thus be (& - z>2)*, and the
time taken is therefore
where /x is the distance AMr
Thus the retardation in time of the former beam relative
to the latter when they are brought together again is
i O *> X O
L^2 - v1 (c2 -
If the apparatus is now rotated so that AMj comes into
the direction of the light, ^ and /2 change places, and the
retardation is altered to
Thus the change in the retardation is
2^+H^-(^W
&
17. ALTERNATIVE EXPLANATION.
The following explanation, alternative to that given above, together
with fig. 3 which illustrates the absolute path of the rays which interfere,
may perhaps make the theory of the experiment a little clearer.
2*
20 RELATIVITY AND THE ELECTRON THEORY
At a certain moment t3 let A3 be the position of the moving plate
and let us consider the paths of the two parts of the beam which are
reunited at that moment. Let Aj be the position of the plate at the
instant A at which that element of disturbance leaves Aj which, travelling
by the path A1M1A3, arrives at A3 at time /3. Then the time t\ is found
thus:—
AA. = v(fs -
+
giving
or
c
FIG. 3.
In the same way let A2 be the position of the plate at the moment
/2 of the emission of the element of disturbance which, travelling by the
path A.iM2A3, arrives at A3 at time /3.
Then A3A2 = i/(/3 - /2),
A2M2 = C(T - /2),
M2A3 = c(t% - r),
where r is the moment of reflection.
Also /2 is the distance of M2 from the position of the plate at time T.
Thifs : — /2 — A2M2 - v(r - 4)
= (c _ ^) (r _ /2),
and similarly /3 = (c + v) (tz - r),
gvng
C - V
/2 ^t_t
+ "v
or
- V*
THE ORIGIN OF THE PRINCIPLE
21
Thus the disturbances which are united at A3 at time /3 do not leave
the plate A simultaneously, but at instants separated by an interval
(^ - -z/2) c* - V*
and therefore will differ in phase by this amount, exactly as obtained above.
The question to be answered by the experiment is whether this difference
of phase will be altered when the apparatus is turned round.
Looking at the theory from this point of view it becomes necessary to
consider whether the directions of the reunited parts of the beam will
necessarily be the same. This involves the question of the reflection of
a ray of light at a moving mirror. Let us consider this by means of
Huygens' principle.
Let AB be a reflector placed at any angle a with SS', and moving with
velocity v in the direction SS'. Let AX be a plane wave-front incident
on the mirror at A at a certain instant ; at a small time r later, the un-
reflected portion of this will have advanced a distance CT, being now part
of the line X'N. In the same time the reflector has advanced a distance
AA' = vr ; so that, drawing A'B' parallel
to AB to meet X'N in C, C is now the
point of incidence of the wave-front on
the reflector. Thus the reflection is
exactly that which would take place at
a fixed mirror in the position AC.
In the same way, considering a
ray incident on the other side of AB,
moving in the opposite direction, we
find that A'C' is the direction of the
equivalent fixed mirror.
Thus if the mirror is set at an
angle of 45° with AS' the direction of the
reflected part of the beam at the first
incidence will be AMi where the angle
XAMX = zC'AC,
and the direction of the transmitted ray after its final reflection will be
A'Z where
YA'Z = 2C'A'C.
The condition that the reunited rays shall be parallel is that
XAMj = YA'Z,
that is that CAC' = C'A'C,
and this is satisfied since the right-angled triangle ANC' is isosceles.
1 8. THE RESULT OF THE EXPERIMENT.
Interpreted according to this calculation, if the velocity of
22 RELATIVITY AND THE ELECTRON THEORY
the earth relative to the aether had been as much as a quarter of
the velocity of the earth in its orbit, there would have been a
displacement of the fringes produced by the interference of
the two beams of such magnitude that it could not have
escaped observation with the apparatus used. But no trace
of such a displacement was found.
Morley and Miller, repeating the experiment in 1905, with
still greater refinement, also came to the conclusion that there
was no displacement, though they could have observed one
due to one-tenth of the velocity of the earth.
19. THE FlTZGERALD-LORENTZ CONTRACTION.
If, as we seem bound to, we accept the above results as
showing the theory which has been given to be inadequate,
the hope of determining v by this means vanishes, and the
difficulty remains of reconciling the null result with the hypo-
thesis of an aether which is not convected along with the optical
system by the earth.
FitzGerald l threw out a suggestion that if the aether can
percolate through matter, it may affect the apparatus, and
change its dimensions when it is rotated. Such a suggestion
had become feasible in view of the adoption of the electro-
magnetic theory of light, and the growing knowledge of the
electrical relations of matter.2
If such an effect is to nullify the change in the retardation,
we must have, if /'p /'2 are the changed lengths of AMX, AM2,
This is satisfied if
that is, if either arm contracts in the ratio I : (i - vl\c^ when
turned into the direction of motion, as compared with its
length when at right angles to this direction.
iSee O. Lodge, "Aberration Problems," "Phil. Trans.," 184* (1893), P.
727 ; also Presidential Address to the British Association, 1913.
2 E.g. Maxwell's law connecting the index of refraction with the dielectric
constant.
THE ORIGIN OF THE PRINCIPLE 23
FitzGerald's suggestion was not carried further at the time,
and it was left to Lorentz to make it indendently,1 and to
show at the same time some plausible reason why a contrac-
tion of exactly this amount might be expected.
The theory to which Lorentz was led in his investigations
into the optical behaviour of moving bodies is so fundamental
to the present subject, and is, if not always in its original
form, so generally accepted, that some account of it must now
be given.
20. THE NULL RESULT INDEPENDENT OF THE MATERIAL
CONSTITUTING THE APPARATUS.
Before this is done, however, we may note a fact of great
importance for the general significance of the Michelson-
Morley experiment. As first carried out, the whole apparatus
was mounted on a sandstone block which floated in mercury.
In the repetition by Morley and Miller, the distance between
the mirrors was intentionally maintained by wooden rods.
Thus if the null effect is to be explained by FitzGerald's
suggestion, the contraction of the right amount must auto-
matically take place in two such different materials as sand-
stone and pine. It is difficult to think of this as a mere
coincidence, so that we are naturally led to think that if we
accept the contraction hypothesis, whatever explanation of it
we may adopt, it must be one that is universal, and inherent
in the constitution of matter, even down to the structure of
the chemical atom.
21. THE INFLUENCE OF THE MICHELSON-MORLEY EXPERI-
MENT ON THE DEVELOPMENT OF ELECTRICAL THEORY.
The contraction hypothesis of FitzGerald is a second in-
stance of an ad hoc addition to existing theory for the express
purpose of making it fit the facts. The first instance, cited
above, was Fresnel's hypothesis of a convection-coeffic ent.
But it was not possible for the matter to rest at this point.
Why should matter contract to exactly the required extent,
1 " Versuch einer Theorie der Elektrischen und optischen Erscheinungen in
bewegten KOrpern," Leiden, 1895.
24 RELATIVITY AND THE ELECTRON THEORY
and that, too, in the case of two such different materials as
stone and wood? The necessity of answering this question
supplied a direct and powerful stimulus to the construction of
a theory of matter in which the suggested change in the length
of a body is no longer an arbitrary hypothesis, but a necessary
consequence.
The result of these efforts was the electron theory of
Lorentz and Larmor. In this theory an ideal picture of matter
in general is constructed. The special properties character-
istic of the particular chemical constitutions of bodies are
eliminated from the consideration at the outset ; the general
laws which govern them are supposed to be common to all
kinds of matter. All matter is conceived to be composed of
electrons, that is, of small nuclei which are indivisible, and,
though this is not absolutely necessary, of identical nature for
all kinds of matter. As far as possible the exact nature of
these electrons, their size and shape, and the way in which the
electric charge is distributed on them, is left out of considera-
tion. These electrons are in motion within the body which
they constitute, and set up an electro-magnetic field accord-
ing to the laws which were first completely formulated by
Maxwell.
This field in its turn is supposed to determine the way in
which the electrons move. But the general laws which deter-
mine the field when the motions of the electrons are given in
advance are not sufficient in themselves to do this.1 To
supplement them some further assumption has to be made as
to the nature of the reaction of the aether on the electron.
We may see at once how the desire to explain the auto-
matic contraction of a body when set in motion through the
aether regulated the choice between the various hypotheses
that might present themselves to the mind. Lorentz showed
that all that was necessary beyond the equations of Maxwell
to determine the motion of an electron was to assume (i) that
its exact size and shape, or, more precisely, the exact distribu-
tion of the electric charge, should be given, and (ii) that the
whole inertia or kinetic reaction of the electron was due to
1 For a fuller discussion, see Chapter IV, pp. 51 ff.
THE ORIGIN OF THE PRINCIPLE 25
the electric field which by its motion it was setting up in the
aether. This second assumption is known as the * hypothesis
of purely electro = magnetic inertia ' ; it was shown to be a
reasonable one by the results of the experiments of Kaufmann
in 1901, on the apparent inertia of the free electrons which
are supposed to constitute the cathode rays.
Now what was the assumption which Lorentz found it
desirable to make about the configuration of the electron?
Starting with a desire to prove the reasonableness of the con-
traction-hypothesis as a universal characteristic of all matter,
he was led to assign the same property to the ultimate com-
mon element That is, assuming an electron at rest to have
a spherical configuration, an electron moving with velocity
v is supposed to have a spheroidal shape, the whole electron
being squeezed together along the direction of the velocity in
the ratio (i - v*/<*y : I.
The reason for this contraction of the electron is not con-
sidered ; neither are the agencies which hold the electron to-
gether ; these are eliminated by means of the purely kinematic
assumption. But granting it to exist, Lorentz was able to
show l considerable reason for anticipating that a body con-
stituted according to his scheme would automatically undergo
the required contraction when set in motion.
This, put rather bluntly, is the general result of the work
of Lorentz. Of course, there was actually more support for
this conception of the electron than the mere fact that it
furnished an explanation of the hypothetical contraction.
Such, for instance, was the fact that it also gave an explana-
tion of the failure of the experiments of Rayleigh and Brace
which sought to find an evidence of double refraction in an
isotropic transparent body when set in motion through the
aether.2 But this does not alter the fact that the suggestion
of the contracting electron is in reality only throwing the
mystery a stage further back. If we accept the suggestion, it
1 Lorentz, " Proc. Roy. Soc.," Amst., 1904 ; also, Larmor, " ^Ether and
Matter," 1900. Larmor, on the ground of the theory developed here, demurred
to Lord Rayleigh's expectation of any effect (Brit. Ass., Belf., 1902).
2 See below, p. 56.
26 RELATIVITY AND THE ELECTRON THEORY
will not be long before we go on to ask the further question,
* Why is this so ? ' This can only be answered by an analysis
of the construction of the electron, in terms of its parts, so
that we should at once be carried beyond the stage at which
the electron is an ultimate element in our thought. At this
point, therefore, the assumption must be left until something
arises which may throw further light on it, or furnish us with
an alternative.
22. LORENTZ'S ARGUMENT ANTICIPATES THE PRINCIPLE
OF RELATIVITY.
At the present moment it is desired to call attention to the
fact that the hypothesis of the contracting electron is nothing
more than throwing back on the elementary conceptual con-
stituent of matter the very property which it desired to
establish of the objects of experience.
This is in reality an application of what came to be known
in 1905 as the Principle of Relativity: that is, it is a result of
the assumption that the failure of experiments to discover the
motion of the earth relative to the aether is no accident, but is
the result of universal relations inherent in the constitution
of matter. If Lorentz's or Larmor's theory be true, then the
velocity of bodies relative to the aether must for ever remain
unknown.1 This is the point from which the 1905 principle of
relativity starts. But the deduction of consequences from this
point is independent of any constitutive theory of the nature
of matter. Lorentz's theory, as far as it goes, implies the
relativity of all its consequences, but many of those results
can be obtained as an immediate deduction from the hypo-
thesis of relativity apart from the details of the theory. Not
only so, but in regions where the pure electron theory becomes
inapplicable, we may say what laws are consistent with the
relativity of all phenomena and what laws are not.
The law of gravitation, for instance, in the commonly ac-
cepted form, is not consistent with the relativity implied by
the electron theory. If matter were completely constituted
according to that theory, the law of gravitation would come
1 See Chapter IV, p. 55.
THE ORIGIN OF THE PRINCIPLE 27
within the scope of the theory, and we should therefore expect
that, even if we could use gravitational phenomena in our
investigations, we should still be unable to find any evidence
of the earth's motion relative to the aether.
The question is thus suggested : —
(1) Are the observations of astronomy consistent with the
hypothesis of relativity in the sense in which we are now
thinking of it ? Or : Do existing observations give us a
means, if only the proper way can be devised, of determining
the velocity of the earth through the aether ?
(2) If we give up hope of the latter possibility, and grant
that in some way the law of gravitation must be included in
the scope of the principle of relativity, is this because, contrary
to the usual opinion, gravitation is really a product of the
electrical constitution of matter, or is the relativity of phe-
nomena in the special sense of which we are thinking, some-
thing of wider range and more deep-seated than the electron
theory itself?
This fact of gravitation and our incapacity to coordinate
it with the other physical qualities of material bodies consti-
tutes perhaps the most important region in which the theory
of relativity can go beyond the theories of the constitution of
matter. It is just where these theories become insufficient to
give a complete account of the mechanism on which the pro-
perties of matter depend that the Principle of Relativity be-
comes of importance as a supplementary hypothesis. The
examination of the question as to what light it could throw
on the problem of gravitation was no doubt a great stimulus
to the revolutionary work of Einstein, which has now gone far
beyond the Special Principle of 1905.
CHAPTER III.
THE RELATIVITY OF SPACE AND TIME.
23. IN order to arrive at the essential point which Einstein
added in 1905 to the argument of Lorentz, we will now
begin to examine some positive results of the fundamental
assumption that it is impossible to determine the actual
velocity of the earth relative to the aether, conceived after the
manner of Lorentz as a unique medium which is capable of
use as a frame of reference.
It is assumed that the fundamental property of the aether,
that of propagating effects with a definite velocity, the same
at all points and in all directions, is one of the phenomena
with which we are concerned. This implies that we shall
assume that the result of the experiment of Michelson and
Morley is general ; that is, we assume that we are ever to
remain in ignorance of any difference between the velocities
of light in different directions relative to the earth or, more
generally, to any given observer.
Now this means that for all practical purposes we may
at any instant suppose the observer to have any velocity we
choose. But we may not, of course, attribute to him any
arbitrary acceleration ; because, as was said at the outset, it is
not yet assumed that the relativity extends to the concealment
of all motion, but only of a motion which is uniform. For
the present, when we speak of an observer we mean an
observer who is moving uniformly relative to the aether, so
that we may choose a frame of reference in which he is
permanently at rest.
24. ON THE IDEA OF SIMULTANEITY.
As a result of the fact that an unambiguous conception
of time emerges out of the system of dynamics, the idea
28
THE RELATIVITY OF SPACE AND TIME 29
of ' simultaneous events at different places ' came to be
considered as one about which there was no lack of defini-
tion.
If we examine the development of the idea, however, we
see that it arose gradually in some such way as this. In
the first instance, events which were seen by the eye at the
same moment were considered to be simultaneous. But in
reality the simultaneity was not in the events, but in the
observer's consciousness. This is, in fact, the only kind of
absolute coincidence about the meaning of which there is no
doubt. For as soon as the finite velocity of propagation of
light is recognised, it has to be allowed for in estimating
the moments of occurrence of events at a distance from the
observer. Not only this, but the velocity of the observer
must be allowed for if he is thought to be moving.
In doing this the actual observations made are necessarily
of what may be called 'coincidences'. Measurement only
becomes exact when mental judgment of distances and in-
tervals is eliminated. It is the chief aim of the experimenter
to do this as far as possible. If we seek to date some celestial
phenomenon, what we observe is a coincidence, that of a light
impression falling upon some terrestrial object, as the eye. In
allowing for the time of transmission of the light, we enter
the realm of theory, and postulate some law such as that of
the constancy of the velocity of light. If this is all we pos-
tulate, and if we have no other means than that of light
signals for setting up a scale of ' simultaneity ' for events
on the earth, and at distant celestial points, then it will be
seen that we are not able to date the event uniquely.
If the conception of a definite and unique aether be adopted,
then for an observer at rest in it a criterion of simultaneity
can be set up as follows. Let A, B be two points, each of
which is supposed to be at permanent rest in the aether, and
let a ray of light be emitted from A at an instant tlt in the di-
rection AB. Let this ray be reflected back on its arrival at B,
the moment of reflection being called /2. Let the reflected
ray arrive at A again at an instant which we will call /8. Then
on the ordinary view, A and B being at rest in the aether, t
30 RELATIVITY AND THE ELECTRON THEORY
must be considered as simultaneous with the instant midway
between t± and t^. We may write
^ = K^ + « - - - (0
On the other hand, if A and B have a common velocity v
in the direction AB, and the same process of transmission of
a ray of light from A to B and back again is carried out, this
will not be so. For if / represents the constant distance AB
and the instants corresponding to t^ /j, 4 are denoted by
A> ^* ** then we have
/«-:/, --#£-4
*•-'•-//('+*>
These equations lead to
Thus in this case the instant of reflection is not to be con-
sidered as simultaneous with the instant midway between the
instants of the start and return of the signal
Now as long as we have any hope of knowing the velocity
of a given point relative to the aether, this fact is in no sense
confusing. But if we grant the hypothesis of relativity, and
reconcile ourselves to the thought that we shall never know
the actual velocity of the points A and B, then we must also
face the consequence that we can never know which instant at
B may be called, more than any other, * simultaneous ' with a
given instant at A. Or we may say that the phrase ' simul-
taneous events at different points ' has no meaning until the
velocity of those points is stated.
This is one of the main distinctions between the outlook
here suggested and that of the earlier thought We cannot,
granting the fundamental assumption, say that there exists on
physical grounds a unique means of ordering phenomena in
time regardless of their position. If on metaphysical grounds
we desire to think that there is really a unique and absolute
meaning for the time which we measure, yet we must admit
that we have no means of knowing whether we are using it or
not It is exactly the same position to which we are brought
in respect of the aether. We may assert its existence, but we
have no means of identifying it, and the definition of it in
THE RELATIVITY OF SPACE AND TIME 31
terms of its properties determines it only as one of an infinite
number, all of which have identical properties.
25. ON THE MEASUREMENT OF LENGTH AND THE
CONCEPTION OF RIGIDITY.
If it be granted that we must leave the question of the
simultaneity of events an unanswered one, we are also faced
with a difficulty about the notion of the distance between two
given points of a material body. In the ordinary or Newtonian
way of measuring lengths on a moving body, the length of a
given line in the body must be defined as the distance be-
tween the two points of space which are occupied by the ends
of the line at simultaneous instants^ the conception of distances
between the points of space being supposed, like the concep-
tion of absolute intervals of time, to be an a priori one.
But if we grant that we cannot speak without ambiguity of
'simultaneous instants' until we have chosen an arbitrary
body to be the one which is said to be at rest at a particular
moment, then it is clear that we cannot, without further
definition, speak of the length of a given line in a body until
we know what velocity is assigned to it
It is taken for granted, in the classical treatment of the
dynamics of rigid bodies, that the distances between the
various parts of a body are invariable in the motion. This
conception of an ideal rigid body is fundamental, and space is
conceived to be so graduated that a given body occupies a
region of exactly the same size and shape at every instant,
and that this region is independent of the velocity of the
body.
We see now that this conception is breaking down at
various points. First we have the suggestion of FitzGerald
that every body contracts on being set in motion ; then we
have the difficulty of defining what we mean by the region
occupied by the body at a given instant on account of the
difficulty about what we mean by simultaneous positions of dif-
ferent points of the body. Not only this, but we have a further
practical difficulty in the fact that if our conception of metrical
space is based on that of an ideal rigid body this seems to
32 RELATIVITY AND THE ELECTRON THEORY
react on the measurement of time, since we have the apparent
truth that equal distances are traversed by a ray of light in
equal intervals of time.
In order to deal with the apparently confused situation so
created it is necessary to make an entirely new beginning and
to build up afresh our statement of what we mean by the
system of measurement of space and time that we employ.
In the remainder of this chapter we shall see the position
adopted by Einstein in 1905. But his thoroughgoing recon-
struction will be described later.
26. THE CONSTRUCTION OF ALL POSSIBLE SPACE AND
TIME SYSTEMS.
We retain for the present the assumption that we are un-
able to determine a difference in the velocity of light in dif-
ferent directions.1 If we did not make some hypothesis of
this nature to replace the graduation of space by means of a
rigid body and of time by the rotation of the earth, we should
have no defmiteness in our system of time and space at all.2
We can order the events which happen in the universe in an
infinite number of ways. Suppose, for instance, that we as-
sociate with any point of space the quantities (ar, y, z), and
with any instant of time at that point the quantity /. If we
take four quantities (x ,y , z\ /), defined as any arbitrary func-
tions of (x, yy z, /), then with any occurrence is associated a set
of quantities (x, y, z, /), and consequently a set of quantities
(x ', y, z\ /). Events may be described, as far as their order
is concerned, by relations between these quantities. For
example, the motion of a moving point may be described
either by saying how (x, y, z) vary as t varies, or how (x ,y\ #')
vary with /. Who shall say, or how shall we know, whether
the former or the latter is the proper set of quantities associated
with our resolution of the history of the physical universe into
space and time sequences.
It is only when we begin to speak of the form of the re-
lations which hold between the motions of different points that
1 Actually this is given up in the new theory.
2 This is, in fact, the case in the general Principle of Relativity.
THE RELATIVITY OF SPACE AND TIME 33
the choice of the variables which are used to describe the
motions is in any way limited. The Newtonian form of the
laws of motion limits us, not to a single set of time and space
co-ordinates, but to a group of an infinite number, the relation
between any one set (x^ ylt #x) and any other (>2, jj/2, #2) being
of the form
x^ = x^ - utv /2 = yl - vtv *,»*!- wt^ ^ = tY
A fixed point in the second set is a point which in the first
set satisfies the relations
x\ ~ ut\ = const-j y\ ~ vt\ — const, ^ - wt^ = const. ;
that^is, it is a point moving with velocity (u, v, w). Here
(u, v, w] are any arbitrary velocities whatever. (See p. 1 1.)
27. THE HYPOTHESIS OF- THE CONSTANCY OF THE
VELOCITY OF LIGHT.
What now is the corresponding limitation imposed by our
attempt to maintain, instead of the Newtonian laws, the fact
of the constant and universal velocity of light ? This limita-
tion is simply that between any two possible sets of space-
time co-ordinates (x^y^ zlt ^), (x^ jj/2, z^ ^2) there must be such
relations that, if a point is moving with velocity c in the first
set, it is also moving with velocity c in the second set.
This may be set down in mathematical form thus : —
If the moving point changes its space co-ordinates by
amounts (f1? 77^ £) in an interval of time TI} and the cor-
responding quantities in the second system of reference are
(£j> ^2> £2) and T2 ; then as a consequence of the equation
fz2 + %2 + &' - <V
we must have
&2 + %2 + ti - ^v.
Now, it is a purely mathematical problem to find out for what
kind of relation between the two sets of co-ordinates this con-
sequence can and must arise ; and it is a problem which is
capable of complete solution. The complete solution gives
us, of course, certain obvious changes of co-ordinates which we
need not discuss ; namely (i) the addition of mere constants
to x, y, 2, and /, and (ii) a uniform change in the scale of all
four co-ordinates, which amounts to nothing more than a
3
34 RELATIVITY AND THE ELECTRON THEORY
multiplication of the units of space and time by the same
quantity, a process which clearly will not alter the measure of
the velocity of a moving point. There is also the obvious
possibility of turning the space frame of reference round into
any other angular position.
But apart from these there is still an infinite number of
ways of changing the co-ordinates. Of these there is an in-
finite number which are linear transformations,1 that is, in which
each of the new set of co-ordinates can be derived from those
of the old set by taking sums of constant multiples of the four
co-ordinates of that set. We may show by a simple algebraical
calculation that these can all be obtained from a change of
variables of the following simple type : —
*2 = £(*! - vt^y* = j/i, ^2 = *lf *, = £(/! - w^) . (A)
Here /8 = (i - z/2/^2) ~* where v is arbitrary.
Suppose now that we consider a point which is at rest in
the system of co-ordinates (x^ yz, #2, /2), that is a point for which
the space co-ordinates (ar2, j2, #2) have fixed values. Then
applying the relation (A) we find that for this point
(*i - vtj, ^i and XT,
have constant values, that is, the point has a velocity v
parallel to the axis of xlt if (x^ yly zlt /1}) are considered to
be its space-time co-ordinates. This velocity v is of arbitrary
magnitude and direction, since the axis of x may be taken in
any direction we please ; so that we may write down a trans-
formation of this type which will cause a point at rest in one
system of co-ordinates to have in the new system any velocity
that we choose to assign.
28. If we put
*2 = j3*j + acfl9 y* = ylt z2 = glt <rfa = yxl + 8ctlt
this gives identically
x? + y? + z? - M? = x£ + yf + g* - c*t*,
provided that /S2 - -y2 = S2 - a2 = i
and a/3 - yd = O.
These conditions lead to
«2 = /32 - i,/ = /32- i,82 = ^
so that, if we put a//3 = - z//r, we have
£2(i - &!<*) = i,
so that the formulae for the change of co-ordinates are as stated.
1 It is possible to generalize all that follows to include the tranformations
which are not linear.
THE RELATIVITY OF SPACE AND TIME 35
29. THE LORENTZ-TRANSFORMATION.
The transformations of type (A) lie at the heart of the
history and early developments of the Principle of Relativity.
They are commonly known as Lorentz-Transformations.
They were not first arrived at in the way which has been
given above. Lorentz was the first to see their significance
for the theory of the optical and electrical effects in moving
bodies, though he neglected the square of v\c, and so took
y8 = I. His results only professed to be accurate as far as
the first order in vie. The approximation was carried a stage
further by Larmor,1 who showed by the use of the above vari-
ables how the second order effects of motion through the
aether would be concealed. Einstein was the first to use the
transformations in the significance which we have seen them
to have.
What we have to do now is to see whether the facts of
physical observation are in agreement with the suggestion that
any two systems of measurement of space and time which are
related to one another by a Lorentz-transformation are equally
valid. We have seen that it is consistent with the facts
known of the propagation of light through free space. It
remains to be seen whether the known facts of electro-
magnetism and dynamics (including observational astronomy)
can be fitted into this framework of space and time without
contradiction ; these being taken as representing the exact
branches of physical science. If this can be done, the result
will be that all the facts at our disposal are unable to
distinguish between these sets of co-ordinates as valid or
convenient measures of space and time.
30. ON THE DIMENSIONS OF A MOVING BODY.
We are now able to express definitely what is to be as-
sumed of the dimensions of a moving body under the funda-
mental hypothesis that we have made.
The ' length of any line in a body ' is, as was said above,
defined to be the distance between two points of space which
are occupied simultaneously by the ends of the line. Let us
1 " ^Ether and Matter," Cambridge, 1900, pp. 173-7.
3*
36 RELATIVITY AND THE ELECTRON THEORY
suppose that we have a body which is at rest in the system
of co-ordinates (x, y, z, t). Let the time-space co-ordinates
for two particular points of the body be (x^ y^ z^ ^) and
(x^ y<b zto ^)> and let tne corresponding quantities in another
system of reference be denoted by the addition of an accent.
Then applying the equation (A) twice and subtracting we
have
*i - »(*2 - yl,- • (3)
** - *i ~ /% - ^- v (^- x^} . (4)
The last equation shows that if we take simultaneous posi-
tions of the two points in the second system by putting /2'
equal to t^, the corresponding times in the first system differ
by an amount given by
0 = *2 - *i - V (** - X^'
But since the body is at rest in the first system, ^ and x% are
independent of ^ and t^
Putting (£/ - //) equal to the value given above, (4), we
obtain from (3) for the difference between the simultaneous
values of x^ and x^ the value
_
that is the apparent length of a body in the direction of the
velocity which it is conceived to have, is less than the apparent
length when it is conceived to be at rest, in the ratio
I : (I - z/2/£-2)% the dimensions at right angles to the direction
of the velocity being the same.
31. Is THE FITZGERALD CONTRACTION A REAL ONE?
Naturally there arises in the mind the question whether
this apparent contraction is in any sense a real one. To
meet this one or two suggestions may be given. Lorentz,
working always with the conception of a unique aether, insists
that the contraction is a real one, and not, as the position
above outlined may seem to imply, a purely subjective change
in the system of measurement
But the question which must be asked is, " What is meant by
a real contraction ? " In what sense is it possible to speak of
THE RELATIVITY OF SPACE AND TIME 37
a contraction at all? It must be a contraction relative to
something, something which is conceived to have a permanent
configuration. This something may be taken to be the con-
ceptual frame of reference, and so long as the frame of reference
is a unique one the question as to the existence is one with a
meaning. But until the frame is defined this is not so.
The position taken then in regard to the contraction of the
apparatus of Michelson's experiment when rotated relative to
the earth is this. Let us assume that, when we take a frame of
reference in which the apparatus is at rest, the configuration of
the apparatus is unchanged when it is turned round so slowly
that the effect of centrifugal forces in straining the body may be
neglected, as well as the effect due to the altered strain arising
from the change in position relative to the gravitational field
of the earth. That is, we assume, in accordance with the
Principle of Relativity, that an observer on the earth will
observe no effect on the apparatus as a consequence of the
motion of the earth. This being so, the conclusion drawn
from the fundamental hypothesis of the principle by the an-
alysis given above is that, if we refer everything to a frame of
reference relative to which the earth is moving, there will
actually be a contraction of the apparatus as measured by
the co-ordinates of this frame.
But, of course, this real contraction will not be capable of
detection ; for in order to measure it physically it is necessary
to lay alongside of the apparatus another comparison body,
and to allow this body to remain alongside and to share the
displacement of the apparatus. It will therefore share also in
the universal contraction, and the measurements will remain
as before.
32. CRITICISMS OF THE USE OF LIGHT SIGNALS.
The objection is often raised against the argument of the
preceding sections that the introduction of this fundamental
assumption of the constant velocity of light for the purpose
of controlling the time standards at different places is in reality
a very artificial proceeding. This objection should be con-
sidered carefully before going any further.
38 RELATIVITY AND THE ELECTRON THEORY
The first question that may be asked is, " H"ave we any
right to suppose that there is no means of communicating
between distant points other than, and independent of, the
transmission of light. If such a means existed it might be
used to set up a standard of simultaneity which is different
from that given by the equation (2), p. 30."
It used often to be suggested, for instance, that gravitation
is propagated instantaneously, or, at any rate, with a velocity
enormously greater than that of light. In view, however, of
the new theory of gravitation to be described later it will not
be worth while discussing this here. The whole of the present
argument must be taken as preparing the way for the broader
outlook.
A second form of the same objection is that in practice
our measure of time is controlled by the dynamical motion of
the earth. This amounts, as a matter of fact, to saying that
the time which we use in practice is the uniquely defined time
of the Newtonian dynamics. But, as we shall see later, the
classical dynamics cannot be considered any longer as absolute
in its accuracy. This is obvious merely from the fact that we
know that moving charged bodies do not obey the Newtonian
laws of motion ; the deviation has been actually observed in
the case of the cathode rays and the /3-rays. We shall hardly
expect therefore, that with the modern view that matter is
largely, if not wholly, of electrical constitution, the dynamics
of material bodies will conform exactly to the simple scheme
propounded by Newton. We are almost prepared, in fact, to
believe that the exact dynamics of matter will be a very com-
plicated matter. Abraham, for example, dealing with the
mechanics of the electron, obtains a formula * for the apparent
mass of a single electron which makes it appear almost hope-
less to think of further progress in the direction of evaluating
the mass of a much more complicated system.
Now, as we shall see later2 just at this point the Prin-
ciple of Relativity was able to step in and indicate a general
modification of the usual formulae of dynamics which renders
1 " Theorie der Elektrizitat," vol. ii, 2nd edition, Leipzig, 1908, p. 180.
2 Chapter V.
THE RELATIVITY OF SPACE AND TIME 39
them consistent with the standards of time and space set up
in the preceding sections. The modifications required are
everywhere of the order of the square or some higher power
of the ratio of the velocity of the body to that of light, and
if these quantities are neglected the formulae reduce to the
ordinary Newtonian form.
These modifications were found to be in strict accordance
with the direct evidence available from experiments on the
motion of the free negative electrons in the cathode and
/3-rays,1 which at present are the only moving systems in
which the velocities are sufficiently great to render the devia-
tion from Newtonian mechanics appreciable.
Further, there is strong indirect support in the null results
of the experiments of Rayleigh and Brace, Trouton and Noble,
which cannot be explained by a combination of existing
electrical theory with Newtonian dynamics.2
The objections raised are therefore such as can only be
maintained by showing either that there are some experimental
facts which are not reconcilable with the hypothesis of con-
stant light velocity, or else by showing that an unnecessary
complication is introduced into our thought by making this
assumption.
33. THE VELOCITY ADDITION FORMULA.
If we consider a moving point which is displaced in time
Bt through distances &tr, ty, §z parallel to the three co-ordinate
axes, the corresponding displacements when measured in the
system (x>y > z, /) can be obtained by applying the equations
(A) twice for the positions at the beginning and end of the
interval S/. On subtraction we get
&r = £(&r - vSf), S/ = fy, &ar' = Sz, S/ = 0(St - vtejc*) (5)
From these, dividing &/, S/, 82' by S/, we get expressions for
the velocities as estimated in the new frame of reference : —
u* - v , 3u u
I - vuxl<*> y i - vuxl<? I -
1 Vide § 52, p. 65.
2 §§47-9 infra.
40 RELATIVITY AND THE ELECTRON THEORY
If we put u equal to zero in these equations, u becomes
equal to ( - v) in the direction of the axis of x.
Conversely since, as we have seen, the formulae (A) can be
reversed by simply changing the sign of vy if u is zero, u is
equal to v in the positive sense.
Thus a point at rest in the first system of co-ordinates and
a point at rest in the second system have a velocity v relative
to one another.
34. RELATIVE VELOCITY.
In the ordinary way of speaking the relative velocity of
two points is defined as the difference of their velocities, this
being the same for all frames of reference. But as a
consequence of the equations (a) this definition would not
give us a determinate value for the relative velocity. We
therefore substitute the following definition : the velocity of a
point P relative to a point Q is the velocity of P in a system of
co-ordinates in which Q is at rest.
Such a system can always be chosen ; for if we take an
arbitrary system, and the velocity of Q in this system is v
then taking the axis of x in the direction of this velocity, and
applying the transformation (a), the velocity of Q in the new
system is, as we have seen, zero.
Thus, for example, if we have three points A, B, C, of
which B has a velocity u relative to A, and C has a velocity
v relative to B in the same direction as &, then the velocity of
C relative to A is not (u + v\ but (u + v)/(l + uvfc2).
35. THE VELOCITY OF LIGHT AS A CRITICAL VELOCITY.
From the relations above, (5), p. 39, we have identically
or &'2(V2 - ^) = W (u2 - **) ; . . (6)
hence if u is less than c, so is u ; that is, if the velocity of a
moving point is less than the velocity of light in any system of
reference, it is also less in any other, provided always that the
velocity v is also less than c.
An example of this may be given in reference to an objection
that has often been raised to the point of view of the Principle
THE RELATIVITY OF SPACE AND TIME 41
of Relativity. It is suggested l that whereas the analysis of
the principle indicates that relative velocities greater than that
of light are physically impossible, yet there is an actual possi-
bility of observing in nature a relative velocity considerably
greater. We may imagine, for instance, two /3-particles shot
out by a radio-active body in opposite directions with veloci-
ties, each equal to nine-tenths of that of light ; and here is an
actual difference of velocity equal to I '8 c. This is quite true,
and the Principle of Relativity has nothing to say against it.
The principle maintains that a velocity greater than c relative
to the observer cannot be observed ; this is in accordance with
the facts in the case in question. Further, the relative velocity
of two bodies, as defined above, cannot be greater than c ;
this also is in agreement with the example chosen ; for if in
the expression (u + v}\(\ + uv/c2) we put both u and v equal to
'9fy we get> not I '8*, but I -8/1 -8 ic, which is certainly less than c.
The position is. not that velocities greater than c are not
conceivable, but that real bodies become illusory in observa-
tion if they are conceived to be moving faster than light. We
shall also see later that the electrical constitution of matter
seems to indicate that a body would suffer dissolution if it
were accelerated so that its velocity were made greater than c.
Not only so, but the energy of a body tends to an infinite
value as its velocity is increased towards c, so that consistently
with our ordinary conceptions of energy, it would not be pos-
sible to set a body in motion from rest with this velocity.
Of course, there is nothing to hinder us if we please from
thinking of bodies which have a real configuration for a velo-
city which is greater than c ; but such bodies would suffer dis-
solution if they were caused to move with a velocity less than
c, and an infinite amount of energy would be used in the re-
tardation.
36. THE FIZEAU EXPERIMENT AND FRESNEL'S
CONVECTION-COEFFICIENT.
We may now consider the way in which the above con-
siderations give an account of the result of Fizeau's experiment.
1 Vide Soddy, " Nature," November, 1913.
42 RELATIVITY AND THE ELECTRON THEORY
We suppose that a light disturbance travels through a re-
fracting substance with a relative velocity c/p ; that is, this
is the velocity in a frame of reference in which the refracting
material is at rest. Suppose now that we change to a frame
of reference in which the material has a velocity v in the
direction of propagation of the light.
Then, applying the addition equation, the velocity of the
light disturbance in this frame of reference is
e.+v
Thus, neglecting z;2, we arrive at once at the convection-
coefficient (I — yLT2).
In the repetition of Fizeau's experiment by Michelson and
Morley the figure arrived at for this coefficient was '442 -f 'O2.1
Taking the known value of //, for sodium light the theoretical
value is "438. Thus the agreement is satisfactory enough.
37. THE DOPPLER EFFECT.
In the same way, we may give also a simple account of
the Doppler effect.
If light, of period T to a given observer, is travelling along
the axis of.tr, any component of the disturbance maybe taken
as proportional to a harmonic function
sin 1 — (t-.
a}
where x and t are space-time co-ordinates jin which the observer
is at rest.
If now we change to a system in which the observer has
velocity ( - v) by putting
*=£(*•'+ vf), t = p (/ + vxl?)
we have
1 " Amer. Journ. of Sci.," 31, 1885. The figure given in this paper is -434, but
there appears to be an arithmetical error, the data given leading to -442.
THE RELATIVITY OF SPACE AND TIME 43
where —,
neglecting
Here r' is the period as seen by an observer at rest in the
new frame of reference. The result indicates that the effect
of a motion with velocity v opposite to that of the light
shortens the apparent period in the ratio (i — v\c)\l.
38. THE CONVECTION -COEFFICIENT IN A DISPERSIVE
MEDIUM.
In treating of the convection-coefficient in § 36, yu, was
defined by the fact that c/p was the velocity of the disturbance
when the medium was treated as stationary. In a dispersive
medium //, is a function of the period of the light. The value
to be taken in the formula obtained is clearly that cor-
responding to the period r as seen by the observer to whom
the medium is stationary. But the period of the light as
observed in Fizeau's experiment is T', that which is apparent
to the observer relative to whom the medium is moving with
velocity v.
By the same method as in the last section we obtain
T(I .
CJllJ
If fj! is the index corresponding to r
> / / ^u,
p - ^ = (T _ T) r
c)r
V/JLT TH/J,
"
-ru C C ( VT (>/i\
Thus - = - ( I - - - V1 ).
/JL IJL \ c OT/
44
RELATIVITY AND THE ELECTRON THEORY
Hence f + v (l - 1\ = 4 + v(i - L
LL \ U?/ LL \ U?
In the last term we may neglect the difference between p
and //,'.
The additional term in the coefficient of v increases the
theoretical value of the convection-coefficient in the case ex-
amined by Michelson and Morley to '451, but this is still well
within the limits indicated by their experiments, namely,
•442 _+ -02. Zeeman l has more recently performed some
further very careful experiments with light of different colours
with a view to confirming the formula last given. He has
succeeded in showing a dispersive effect agreeing very closely
with the above formula.
In the following table, the values of the convection-coeffi-
cient are given ; in the first column, according to Fresnel's
formula (eF), in the second, according to the formula here given
(eL), and, in the last, the value experimentally found.
Wave-length in A°
«F
«L
6EXP.
4500
0*443
0-464
0-465
4580
0-442
0-463
0-463
546i
0-439
OH54
0-45I
6870
o'435
0*447
0-447
39. THE CONTRACTION IN VOLUME OF A MOVING BODY.
It follows at once from the formula obtained above that if
a body, all of whose points have the same velocity, has a
volume V0 when it is considered to be at rest, and volumes
V, V when its velocities are u and u respectively, then
V = V0(i - u2fc^, V = V0(i -tf'V).*
Now if u is the velocity obtained from the velocity u by the
equations (A), we find on substitution from the addition for-
mula that
"Amst. Proc." x'vu. (1914), p. 445; xvm. (1916), p. 398 and p. 1240.
THE RELATIVITY OF SPACE AND TIME 45
which leads to ^
so that V/V = /3(l - vux\c^ . fr)
This equation gives us the contraction in a volume which is
not at rest in either frame of reference. It will be of import-
ance when we come to consider later whether the measure of
an electric charge is an absolute or relative magnitude (p. 49).
CHAPTER IV.
THE RELATIVITY OF THE ELECTRO-MAGNETIC
VECTORS.
40. THE DEFINITIONS OF ELECTRIC AND MAGNETIC
INTENSITIES.
IN the ordinary treatment of electrical theory the definitions
given of the electric and magnetic intensities at a point are
somewhat as follows : —
The ' electric intensity ' is the force which would be
exerted per unit charge on a small charged body placed at
rest at the point.
The ' magnetic intensity ' is definable from the law that
the force per unit charge on a small moving charged body
is equal to the sum of the electric intensity and the vector
product l of the velocity and the magnetic intensity.
An alternative definition of the magnetic intensity is often
given as the force per unit pole on a magnetic pole placed
at the point. As this introduces the non-existent isolated
magnetic pole, it will be as well to confine the discussion to
the former definition.
We see that, apart from the fact that the measurement
introduces a reference to the mechanical category of ' force,'
the definitions involve the knowledge of the velocity of the
small test body. But we have already learned to look upon
the velocity of a body as a quantity which, physically, is so
far indeterminate. To assign a definite value to it we have
to decide which of all possible frames of reference we will
adopt. Thus the definitions of the electric and magnetic . in-
tensities are relative ones ; their values are ' relative ' in the
same sense as the measures of space and time.
We need therefore to add to the relation (A) (p. 34), a re-
lation between the values assigned to the electric and magnetic
1 See note on vectors, p. I.
46
RELATIVITY OF ELECTRO-MAGNETIC VECTORS 47
vectors in two different systems of reference. We cannot
obtain this from the definitions given just now, because they
involve the conception of ' force,' and we have not yet seen
whether this is to be looked upon also as a relative or ab-
solute quantity. We shall see as a matter of fact when we
come to the consideration of dynamical quantities that we are
bound to treat it as relative.
In order therefore to obtain the required relations we
must adopt the same method as that used above in respect of
space and time, and consider the measures of these quantities
in the light of the laws in which they occur, the laws which
express the uniformities which we believe on experimental
grounds to hold between electrical phenomena.
These laws are the equations of the electro-magnetic field.
We will take them in the form adopted by Lorentz,1 namely,
i/3e
Is
= curl h . . I
o = div h II
_ i — = curl e . • III
c <>/
p = dive e . . IV
where e and h are the electric and magnetic intensities, and
p is the density of charge.
The fifth equation of Lorentz's theory,
F = e + [uh]/r
where F is the * mechanical force ' per unit of charge moving
with velocity u, embodies the definitions given above, and
supplies us with a means of connecting up with mechanical
theory, but gives us no criterion about the purely electrical
magnitudes, the mechanical categories now occupying a de-
rivative place.
41. THE INVARIANT FORM OF THE FUNDAMENTAL
EQUATIONS.
The first thing that may be noticed about the form of the
equations I-IV, is that if we change the scale on which we
1 The units here used are those of Lorentz. See " Theory of Electrons,"
Leipzig, 1909, p. 5.
48
RELATIVITY AND THE ELECTRON THEORY
measure e and h, we alter the scale on which we measure p in
the same ratio. Thus it becomes possible to limit the con-
sideration to such changes of the variables as leave the charge
carried by any particular element of volume unaltered.
The fundamental hypothesis of electrical theory which takes
the place of the Newtonian conception of 'conservation of
mass ' is that of ' conservation of electric charge '. We pro-
ceed to examine, therefore, whether it is possible to make such
changes in the variables e and h, when we change the space-
time co-ordinates according to the transformation (A), that the
form of the equations I -I V may be unaltered and at the same
time the charge carried by any particular element of volume may
be unaltered. The Principle of Relativity sprang from the
discovery that it is possible to satisfy these requirements.1
The following are the equations which give us the neces-
sary changes in the vectors e and h : —
- vhz\c\ ez = f$(ez + vhy\c]
+ vez\c), hz = $(hx - vey\c)
If we take the equations of Lorentz and make the changes
of variables given by (A) and (B), we arrive directly at the
equations
. II'
. Ill'
IV
ev =
o = div' h' .
I^-curfrf.
c ^
= div' e' .
where u' is a velocity whose components are
— V
* I -«,„/«»
and p is defined by the equation
p = £p(l - vux\<*) .-
1 A. Einstein, " Ann. der Phys.," 17, 1905, pp. 891-7.
(a)
(C)
RELATIVITY OF ELECTRO-MAGNETIC VECTORS 49
In order to see the significance of these quantities u' and p,
we must recall one or two of the immediate consequences of
the change in the manner in which space is measured from
the co-ordinates (x, y, z, t) to (#•', y, z\ /).
We have seen, (p. 39) that the equations (a) are exactly
those which connect the estimates of the velocities of a mov-
ing point as measured in the two systems respectively.
We have also seen, (7), p. 45, that if SV is the volume of a
small region of a body moving with velocity u as measured
in (x, y, z, t\ and SV, u' are the corresponding volume and
velocity as measured in (x , y, z , t'} then
8V = 8V/£(i - vux\c^.
If we put this equation and (C) together we obtain
p'SV' = pSV.
Thus the equation (C) is exactly that which is required to
express that the charges in the corresponding elements of volume
are the same.
The equations (A), (B), (C) then effect an exact transforma-
tion of the fundamental equations I-IV into equations of the
same form, and are consistent with a unique value being
assigned to any element of charge. We say that ' the charge
is an invariant*.
42. THE CONSEQUENCE OF THIS TRANSFORMATION.
The result of the exactness of this transformation is that,
so long as we are dealing with phenomena of which the sole
laws are the field equations in the form which we have taken,
we shall not be able to discriminate between one set of space-
time co-ordinates and another, as the only one relative to which
the laws have the required form. In other words, remembering
that as we know it, the aether is simply the frame of reference
relative to which these equations hold, as long as we are
dealing with phenomena which are entirely governed by these
laws, we can never expect to identify the aether. Unless we can
light on some measurable phenomena which are not subject
to these field equations alone, the aether remains an unknown
frame of reference, exactly as the frame of reference for the
laws of dynamics is indeterminate.
4
So RELATIVITY AND THE ELECTRON THEORY
Lorentz remarks that it is really a matter of personal
predisposition whether this leads a physicist to deny the
existence of the aether or to the more agnostic position
which merely says that "the aether is real but will always
be hidden from our knowledge". This may be so, but
few to whom the means of the propagation of electrical effects,
and, indeed, the whole structure of matter is a question of
the first interest, will feel content to take either of these posi-
tions. If the aether, in the ordinary sense of a stationary
medium, does not exist, what is it that conveys the waves of
light through space ? If it does exist, is it a pure accident
that its laws are of such a form that it must for ever remain
concealed ; or is it that we have naturally lighted first on
those aspects of its working which, as being independent of
motion through it, would be most readily found by an observer
who did not know his velocity through it ? 1
In any case, the impression made by the situation is that
there is a break in our knowledge. The Principle of
Relativity represents an attempt to see what are the con-
clusions to be drawn from an admission of our inability to
identify the aether, in the hope that it may throw light on
the question where we are to look for the way out of this
difficulty. In order to see at what point the Principle is an
independent hypothesis, something which is not contained
in a purely mathematical result arising out of the particular
form of a set of equations, we must go a little further, and
consider the way in which the electron theory professes to
give an account of the structure of matter.
43. THE METHOD OF COINCIDENCES IN EXPERIMENTAL
OBSERVATION.
The fact that we cannot have a unique measure of the elec-
tric and magnetic intensities may perhaps be a little more readily
accepted if we consider that in practice we never really profess
to measure them directly. When we, for example, measure
an electric potential by means of an electrometer, the only
actual observation is one of a configuration of relative equili-
1 This is suggestive of Prof. Eddington's observation in regard to the
general principle of relativity — " Space, Time, and Gravitation, p. 201 : " The
mind has but regained from nature that which the mind has put into nature."
RELATIVITY OF ELECTRO-MAGNETIC VECTORS 51
brium. We observe the coincidence of a pointer or of a spot
of light with a certain mark on a scale. Only by eliminating
the uncertain factor of human judgment of an interval of
space and time is it possible to make accurate measurements.
We make our instruments do all the observing for us. Then
if we want to interpret the indications of the instrument in
terms of the values of the strength of an electric field, or of an
electric current or the like, we have to do so by means of
exactly those theoretical equations which we have seen to
have a purely relative significance.
Now it is a fundamental property of the systems of co-
ordinates that we have been considering, that events which
are simultaneous and coincident in space at a given instant
in a given set of co-ordinates are also simultaneous and coinci-
dent in any other of the sets of co-ordinates. To a given set of
values of (x> y, z, t) there is only one value of each of the co-
ordinates (^',y, z , t'\ and these values do not depend upon any
particular circumstances associated with the place or instant.
Thus if we read an electrometer, the coincidence of pointer
and a certain mark on the scale does not depend on the frame
of reference that we are using. But the significance that we
attach to the observation does do so. The quantitative de-
scription of the state of an electrical system depends on the
frame of reference used ; the nature of this dependence is limited
by the fact that the laws of the field are to be independent of
the choice of the frame of reference, and this means that the
estimates of electric and magnetic force for two different frames
of reference will be related in exactly the way represented
by the equations (B) which are necessary for the preservation
of the form of the laws.
44. THE INCOMPLETENESS OF PURELY ELECTRICAL
SCHEMES OF THE CONSTITUTION OF MATTER.
The foundation of the electron theory of matter is the
scheme olTequations which has been considered in the preced-
ing sections. But there are certain other hypotheses that
have to be added to those general laws.
The first is that of ' the atomic nature of electricity '. The
negative ' electrons ' are supposed to be small discrete nuclei,
4*
52 RELATIVITY AND THE ELECTRON THEORY
of identical structure and charge. It is not necessary here to
go into the evidence for this. The magnitude of these nuclei
is not definitely known ; it is only possible for us to construct
tentative theories as to their exact structure. For certain pur-
poses we may be able to treat them as points, for others we
may need to adopt some provisional conception of their extent
and constitution. For the present so little is known about
the properties of positive electricity that we must be content to
admit that it is by no means certain that a similar statement
of the identity of the elementary quanta of positive electricity
is possible.
The picture which we have of the constitution of a portion
of matter is one in which these electrons are grouped and held
together in a permanent configuration ; not in a static state,
but in a state of motion ; the motion of each electron is gov-
erned by certain laws of which the theoretical equations that
have been already referred to are a partial expression.
As they stand it is impossible for them to be the complete
scheme of laws. For it is easily seen that they are not mathe-
matically sufficient to determine, from a knowledge of the
whole field at any given instant, the way in which the field
and the distribution of charge will vary through all time.
The equations II and IV (p. 45) are not independent of I
and III,1 so that we really have only two vector equations
connecting the three vectors e, h, and u, p being connected
1 From III, since div curl e = o (see note on Vector Analysis, p. i), we have
2- div h = o, showing that at a given point div h has a constant value. If at any
preceding time there was no field, this constant value must be zero. Thus Eqn.
II is deduced from III. From I and IV we obtain
£- (div e) + div (pu) = o,
ot
or 2? + div (pu) = o.
ot
This is what is commonly known as the equation of continuity or of conservation
of electricity. Integrated through a closed volume it gives—
that is, the rate of loss of electricity included in volume is exactly accounted for
by the outward flow over the surface ; there is no creation or destruction of
electricity.
RELATIVITY OF ELECTRO-MAGNETIC VECTORS 53
with u by the condition of * the conservation of electricity/
which requires that the change in the distribution of electricity
takes place entirely by convection.
We cannot therefore pretend that our scheme of the struc-
ture of matter is even theoretically complete until we have
added to the general equations some further condition. It is
just at this point that the structure of the electron has to be
taken into account. If we choose to assume that we may
neglect its dimensions, and treat it as a point charge, this is a
sufficient condition.1 The quantity p is then everywhere zero,
and the solutions of the equations which represent the field
are such as are restricted to possess singularities of a certain
definite type. But as a matter of fact this is not a possible
way of building up a practical theory because it introduces
mathematical difficulties in the very simplification of ignoring
the dimensions of the electron. The specification of the exact
way in which the electric and magnetic forces become infinite
in the neighbourhood of a point-electron is a matter of consider-
able difficulty, and the only way of avoiding this is to assume
the electron to have a structure of finite size, and to assume
some definite relation between the distribution of charge
through its volume, and the quantities e, h, and u. Such a
hypothesis will be purely empirical ; it must be chosen ac-
cording to the indications which we have of facts that are
not explained by the general equations.
So the question arises : " What light is thrown by experi-
ment on the structure of the electron?" This question has
been rather obscured recently by the more tangible one of
the structure of the atom as a group of electrons ; but this
has only become possible because, on any theory of the consti-
tution of the electron, its extreme minuteness does allow of
it being treated for some purposes as a material particle of
definite mass and charge, so that it is amenable to ordinary
mechanical treatment.
It would seem that if the structure of the electron can be
ignored in problems of the atom, it would a fortiori be so in
the discussion of the grouping of atoms to form matter. But
1 As in Larmor's " /Ether and Matter".
54 RELATIVITY AND THE ELECTRON THEORY
in the particular problem which we are considering, namely,
the failure of experiment to determine uniquely the frame of
reference which we call the aether, this is not so. For the
preceding analysis of the fundamental equations has shown us
that this failure arises partly because the equations of the
electric field have a form which is invariant under a certain
transformation. We are thus able to form any number of
pictures of a given field, each using a different frame of re-
ference, and each subject to the same fundamental equations.
But if we wish to extend our picture so as to take in not only
the sethereal field but the whole of the matter with which it
is in interaction, it is necessary that any further hypotheses,
introduced to supplement the equations of the field to obtain
a complete scheme for the determination of the configuration
and motion of the various parts of the matter, must also be of
such form that they too are independent of the particular
frame of reference that is adopted.
In order to illustrate this point we may consider a little
more in detail the explanations offered by Lorentz of the
failure of specific experiments, as representing the point of
view of one who thinks in terms of the actual existence of a
unique stationary aether. (Cf. Chapter II.) '
45. THE EXPERIMENT OF MICHELSON AND MORLEY.
We have already seen that the failure of this experiment to
discover any trace of a motion relative to the aether can be
completely accounted for by the hypothesis of a contraction
of the apparatus. The magnitude of this contraction is
exactly that which is apparent when, the body being sup-
posed unaltered, the frame of reference is changed according
to the transformation of Einstein. What reasons are given
for supposing that the body will actually undergo this con-
traction when its velocity relative to a fixed frame of reference
is changed?
The argument, relieved of the mathematical details, is as
follows : When a body is at rest in the aether each electron
within it has a specific motion determined according to
certain laws. If we know these motions completely we can
RELATIVITY OF ELECTRO-MAGNETIC VECTORS 55
by means of the space-time correlation build up a picture of
another system which is moving as a whole with velocity v
relative to the same aether ; in this the motion of any particular
electron is obtained from that of the corresponding electron
in the original body by merely turning the relations between
(xt y, z) and / that express the way in which it moves into
relations connecting (x,y, z} with /. If this is done it is
clear that the configuration of the body as a whole in the new
picture is exactly that of the original body contracted in
the ratio I : (i - z/2/^)*, as on page 36. But, the argument
goes on, the electro-magnetic field of the electrons in the
new body satisfies the same fundamental laws as that of
the original body, and is therefore equally capable of exist-
ing. Thus if these were the only laws in question we should
be fairly justified in concluding that the original body would
automatically change into the one of which we have so formed
a picture when it is set in motion with the velocity v.1
46. INCOMPLETENESS OF THE ARGUMENT FOR THE
FITZGERALD CONTRACTION.
It is clear that there are two gaps in this argument. First,
there is the obvious restriction that it must be assumed that,
provided that the acceleration of the body is sufficiently slow,
so as not to produce mechanical distortion, change of tempera-
ture or other disturbances, there is a unique configuration for
a given grouping of electrons moving as a whole with a given
velocity relative to the aether. This does not seem to offer
any serious difficulty, though it must be remembered that the
external configuration of a material body must be thought of
as a statistical one ; that is, in view of a probable kinetic con-
stitution, to a fixed external configuration will correspond a
multitude of rapidly changing distributions of electrons in the
interior ; but this will not be considered here. The more im-
portant point for discussion is that the correlation which has
been shown to exist in the case of the * field equations/ must
be assumed to hold for all the other relations which play a
part in determining the motions of the electrons.
1 Cf. also " jEther and Matter," p. 176,
56 RELATIVITY AND THE ELECTRON THEORY
It has been remarked that it is enough for the purpose in
hand to supplement the field equations by a hypothetical rela-
tion between the distribution of the charge that constitutes
the electron and its velocity. If this is done, it must be a
relation which maintains its form in the correlated moving
system that we have built up. Since it is a purely geometri-
cal relation it must be a relation which is invariant under, or is
a consequence of, the fundamental transformation (A). Hence
Lorentz's assumption that the electron, which is naturally
thought of as symmetrical round a centre when it is at rest,
is, when in motion, of spheroidal shape, being obtained
from the spherical stationary electron by the simple appli-
cation to it of the FitzGerald contraction. If this is not
assumed, the correlation between the stationary and the mov-
ing system is not perfect, and we have no reason, therefore,
for thinking that the desired contraction of the whole body
will take place.
Thus the whole explanation reduces to the formation
of a conception of the electron, which really embodies the
fact that we are trying to explain. The origin of the con-
traction is left in obscurity. The conception is only arrived
at by assuming the result of the experiment to be, not an
accident arising out of the particular circumstances of the
case, but inherent in the constitution of matter down to its
most minute elements. In other words, the formation of this
conception of the electron is itself a direct application of the
Principle of Relativity.
47. THE EXPERIMENTS OF RAYLEIGH AND BRACE.
If we adopt the conception of the electron that has been
described there is no need to go any further into the details
of any phenomena which are explicable in terms of the motions
of electrons within a body in order to be sure that we shall
not be able to obtain a positive effect due to the motion of
bodies through the aether. For as we have already remarked
more than once, the adoption of a definite relation between
the velocity and the configuration of the electron, so that the
latter is determined uniquely by the former, makes the scheme
RELATIVITY OF ELECTRO-MAGNETIC VECTORS 57
of the laws of the motions of the electrons complete, and the
whole scheme is now a relative one ; the sether is completely
concealed.
It may, however, help to make the consequences of the
conception clearer if we consider specially some of those results
of it ; especially some which directly affect certain experiments
which were devised to detect effects which, it was thought,
would follow from the contraction hypothesis.
It occurred to Lord Rayleigh that a shrinkage of a trans-
parent body in one direction would cause it to cease to be
isotropic, and that therefore there might arise a double re-
fraction of a beam of light traversing it in a direction across
the line of the motion. This was demurred to by Sir Joseph
Larmor when the paper was read in 1902 at the British As-
sociation meeting on the ground of an argument similar to
that given above. But the experiment was carried out again
with extreme refinement by Brace in 1906, the only result
being a complete fulfilment of the prediction of its failure.
The mechanism of refraction as ordinarily conceived is that
the light waves travelling through the body set in vibration
the electrons which go to make up the atoms, and their inertia,
number, and distribution determine the amount of the refrac-
tion. If these are not isotropically disposed the body will
produce a different refraction for different directions of the
components of the electric force in the wave, and so a double
refraction will arise. Now, Lorentz l was able to show that,
starting from the conception of the electron which he had de-
veloped, the effective inertia of an electron is not the same for
all directions. On allowing for this it appears that the in-
equality in the inertia is exactly of that amount which is re-
quired to neutralize the inequality in distribution produced by
the contraction of the body. This, of course, in the light of
what we have said above, is exactly what we should expect.
48. THE EXPERIMENT OF TROUTON AND RANKINE.
In spite of the results of earlier experiments and of the
theory of Lorentz and Larmor it still seemed to some that
1 " Theory of Electrons," p. 217.
58 RELATIVITY AND THE ELECTRON THEORY
there might be need of putting the matter to a further test.
Again reasoning from the FitzGerald contraction as producing
a lack of isotropy in a body, Professor Trouton l suggested that
there might be a difference of electrical conductivity in dif-
ferent directions in a conducting body carried along with the
earth through the aether.
The conductivity of a body like its refracting power
is interpreted in terms of the inertia and distribution of
electrons which move within the body ; and again, if the
difference in apparent inertia for different directions is taken
into account, the effect of the contraction on the distribu-
tion is annulled. The expectation of a positive effect is
therefore unjustified, as we might predict on general grounds
without any calculation of the inertia of the electron at all.
The result of £he experiment again verified this conclusion.
49. THE EXPERIMENT OF TROUTON AND NOBLE.2
In this experiment, which arose out of a suggestion made
by FitzGerald, a parallel plate condenser was suspended with
the plates in a vertical plane and capable of turning about a
vertical axis. An attempt
was made to detect a tend-
ency for the condenser to
set itself in a definite direc-
tion when charged. This
Fwas expected to happen
for the following reason :
If a charged condenser is
in motion through the
FlG- 5- aether, the electric force
between the plates being E, then a magnetic force is set up
in the region between the plates equal to E^sin 0/c, to the
first order in (vjc\ where v is the velocity of the condenser
and 6 is the angle between the normal and the direction of
motion. This magnetic field makes a contribution to the
energy of the moving system proportional to
111 Proc. Roy. Soc. " (A), 8 (igo8), p. 428.
2" Proc. Roy. Soc.," 72 (1903), p. 132.
RELATIVITY OF ELECTRO-MAGNETIC VECTORS 59
But in addition to this the electric force between the plates
is altered by a quantity of the order EV/^2, and so there is a
further change in the energy which is also of the order E2t/2/^.
A calculation gives for the total change of energy the quantity
W = 47rCV2(sin2 0 - $v*/<?
where C is the capacity of the condenser and V the potential
difference. If now the condenser is rotated so as to alter
the angle 6 without altering the charge, work will -have to
be done in changing the energy; this indicates that there
d'W
is a couple acting on the condenser of magnitude —^~
au
that is, - 8-TrCV2 sin# cos# v2/<?. This couple vanishes if
0 is equal either to o or to ^TT ; that is if the condenser is
either normal or parallel to the direction of motion, the
stable position being that in which the energy is a minimum,
that is, when the condenser is normal to the velocity.
The method of observation was to reverse the charge of
the condenser at regular intervals of time equal to half the
time of the natural period of swing of the condenser. By this
means it was hoped that the cumulative effect of the alternat-
ing couple would gradually set up a measureable swing in the
system. The experiment was tried at different times of day
so as to ensure that by no accident the actual velocity of the
earth relative to the aether was normal to the plane of the
condenser ; but no effect could be detected in any case.
If the reasoning whereby we were led to anticipate this
result is correct, the expression for the energy, which is ob-
tained after allowing for the FitzGerald contraction, leads to a
disagreement between the result of experiment and the con-
clusions drawn from applying the ordinary mechanical prin-
ciples. This brings us to the point that was overlooked by
the designers of the experiment. In experiments which take
account of the second order in vjc, the Newtonian mechanics
require correction. This is clear from the case of the electron
that has already been referred to. The apparent inertia being
different in different directions, the force on the electron is
not in the direction of its acceleration. It will clearly be a
matter of some difficulty to derive from a constitutive theory
6o RELATIVITY AND THE ELECTRON THEORY
of matter as composed of electrons of Lorentz's type the
general way in which the dynamics of a rigid body are to be
modified. The only feasible way of finding the kind of correc-
tion required is to start from the hypothesis of relativity in
the new sense, not in the old dynamical sense, and to see what
is the simplest modification that can be made in accordance
with it, just as Lorentz's conception of the electron is the
simplest type of nucleus that can be devised to accord with it.
Some indication of the solution of this problem will be given
in a later chapter, where a fuller discussion of the experiment
will be found. (See p. 8 1.)
CHAPTER V.
MECHANICS AND THE PRINCIPLE OF RELATIVITY.
50. THE TRANSITION FROM ELECTRICAL TO MECHANICAL
THEORY.
IN making the transition from the purely electrical theory that
we have been chiefly considering in the last two chapters to
mechanical theory, it is necessary first to state clearly what is
the connecting-link.
We might follow the method of the critical school which
treats of force as a mere concept, the formation of which is
possible on account of the physical laws of conservation of
mass and momentum. This school of thought begins with
the hypothesis, based on experience, that the accelerations of
two particles which act on one another are in a constant ratio.
This is, in effect, Newton's law of equal and opposite action
and reaction, combined with the conception of a mass-ratio for
two particles which is constant and independent of their veloci-
ties and of any other circumstances. In order to adapt this
hypothesis to the requirements of the Principle of Relativity,
we should have to modify it in such a way that the kinematic
law which we put in its place has a form which remains the
same when we change by a JLorentz transformation from one
set of space-time co-ordinates to another. Having done this,
we might define the force acting on a particle in a convenient
way, and hence obtain the manner in which we conceive the
measure or direction of a force to change when we change
the frame of reference. Then subsequently we might bring
the force, as so defined, into relation with the electrical field
61
62 RELATIVITY AND THE ELECTRON THEORY
by seeking for a quantity in terms of the electro-magnetic
vectors which is subject to the same transformation as that
which has been suggested on kinetic grounds. For any
equation which embodies a relation actually subsisting in
physical phenomena must, according to the Principle of Re-
lativity, have a form which is independent of the particular
frame of reference employed.
It may be as well to point out by an illustration that
some modification is necessary.
The addition equation of velocities which is fundamental in
the relativity kinematics renders the Newtonian form of the
equation of the conservation of momentum no longer possible.
If we consider two particles moving in a straight line under
no forces save their own mutual action, we have the equation
constant.
If we change the frame of reference, by the use of the equa-
tion (a\ p. 39, we have in place of this
«/ + v u.2' + v
m, — - — ; — — o + m9 — - — ; — TO = constant,
1 I + U^vl<? 2 I + UsV/<*
which is of an entirely different form.
It is not easy, without further analytical assistance, to see
in what way the original equation is to be modified in order
to obtain a form which is the same in both frames of reference.
Minkowski devised an analytical method of great power and
elegance by which it can be done very quickly. To this we
shall come in the next chapter. We turn, for the present, to a
means of bridging the gap between electrical and mechanical
theory, which is nearer to that which has been usual in electro-
dynamics, and which is followed by Lorentz, who makes the
transition by introducing the expression for the ' mechanical
force per unit charge,' e + [uh]/£
51. APPLICATION OF THE LORENTZ TRANSFORMATION;
MOTION OF A CHARGED PARTICLE.
We adopt this definition and proceed to consider the
motion of a charged particle. In order to do this it is sufficient
MECHANICS AND THE PRINCIPLE OF RELATIVITY 63
to consider a transformation between two systems, in one of
which the particle is instantaneously at rest. We shall assume
that in this system the acceleration of the particle is given by
q e = m t ,
q being the charge on the particle, f the acceleration, and m
being a definite constant. That is, we assume that for a
particle whose velocity may be neglected, the motion is given
by an equation of the ordinary Newtonian form, m being the
* apparent mass ' when the velocity is zero.
Now, if we take the addition equations
I + *J>f<r p(I + */)\C) P(l + uxv\c*
and assume that u is very small, these give us to the first
order in u
ux' = v + ux(l - v2l<?), Uy = Uylfr «,' = uJ/3.
Now, let us suppose that the velocity u is the velocity com-
municated to the particle from rest in a short time St ; the
corresponding interval of time in the other frame of reference
is to be obtained from the equation
Sf = /3(St + v Sx/c2)
in which we have to put S* equal to zero since the velocity of
the particle is momentarily zero. Thus
M = /3 St.
Hence, dividing the increments of velocity by the correspond-
ing intervals of time, we have for the relation between the
expression for the accelerations in the two systems
But we have also from the transformation of the electric and
magnetic intensities,
Thus the equations of motion become
64 RELATIVITY AND THE ELECTRON THEORY
or, introducing the ' force ' vector F' = (e' + [u'h]/f), since
u' = (v, o, o) we may write these equations
These are the equations which are satisfied by the motion of
the particle relative to the system in which the particle has a
velocity v parallel to the axis of x. They are commonly inter-
preted by saying that the particle has a ' longitudinal mass '
equal to @3m, and a f transverse mass ' equal to f$m ; the word
* mass ' being understood as the ' force per unit of accelera-
tion produced,' and the 'force per unit charge' being de-
fined to be (e + [uh]/£).
As far as the dynamics of the particle are concerned, the
above equations rest on the assumptions, (i) that observations
of the acceleration of a charged particle by an electro-magnetic
field will not furnish us with a means of distinguishing be-
tween the different systems of co-ordinates for which the laws
of purely electrical phenomena preserve their form ; and (ii) that
if the velocity of a particle relative to the frame of reference
is sufficiently small, the law of its motion reduces to the New-
tonian form, the form which has been adopted as the result
of the observation of the motions of bodies whose velocities
relative to the earth are all very small compared with the
velocity of light.
52. THE EXPERIMENTS ON THE DEVIATION OF THE
£-RAYS BY AN ELECTRO-MAGNETIC FIELD.
We may now proceed to consider what light is thrown upon
the equations and assumptions that have been written down
above by the well-known experiments of Kaufmann and others
on the inertia of the negative electrons. In the first form of
these experiments a stream of the /3-rays from a small piece
of a radio-active substance passes through a fine hole in a
screen and impinges on a photographic plate. Electric and
MECHANICS AND THE PRINCIPLE OF RELATIVITY 65
magnetic fields are then set up in a common direction at right
angles to the stream ; it
is then found that, in-
stead of meeting the
screen in a single small
spot, the
duces on
stream pro-
the plate a
curved trace.
The interpretation of
this is as follows. Sup-
posing the rays to con-
sist of a stream of small
charged particles, they
FIG. 6. — Cj, C8, condenser ; P, photographic
plate ; S, S, electro-magnet ; Ra,
radium fluoride.
will be deflected by the electric and magnetic fields. Let us
take the axis of x to be in the direction of motion, and the axis
of y to be in the common direction of the fields ; let the accelera-
tions in the directions of the axes of y and z be fy and fgt the
velocity along the axis of x being v\ then if / is the distance
traversed through the field, and / is the time of transit,
/ = vt\
and the deviations in the directions of the axes of y and z are
y
The measurement of the deviations y and z on the photographic
plate thus gives the values offy/v2 and/Jv^.
Let E, H be the respective intensities of the electric and
magnetic fields. Then, if the equations obtained above are a
correct representation of the facts, we have, since with the given
arrangement the components of F are (o, E, z/H/r),
from which we have
Thus, corresponding to any particular point on the trace of
the ray on the plate, we can obtain an estimate of the velocity
of the particles which strike the plate at that point, the strengths
of the fields being supposed to be known. We are then able
5
66
RELATIVITY AND THE ELECTRON THEORY
to test the correctness of the assumptions that led to the above
equations by means of either equation. The first gives us
q &,
m =" E '
and each factor on the right-hand side is now a quantity of
which we have an estimate. If it be true that all the particles
forming the stream are identical in nature, then the values of
q\m obtained from all the different points of the trace of the
rays on the plate should be the same.
53. THE METHOD OF THE CROSSED FIELDS.
An improvement on this method was introduced by Bestel-
meyer. ' It is now known as the method of ' crossed fields '.
s . s
FIG. 7.
S
The figure shows the essential parts of the apparatus as
used by Bucherer and Wolz. Q and C2 are the plates of a
condenser, consisting of two parallel optically- plane plates of
glass silvered on the inner side, and kept at a fixed very small
distance (about -25 mm.) by pieces of a thin sheet of quartz.
R<z is a grain of radium fluoride from which the /3-rays can
pass between the plates of the condenser to strike the photo-
graphic plate P, which is at a measurable and adjustable dis-
tance from the condenser. The whole of this part of the
apparatus is placed transversely inside a long solenoid, of which
the rectangular section SSSS is shown in the figure. The
electric field between the plates of the condenser and the
MECHANICS AND THE PRINCIPLE OF RELATIVITY 67
magnetic field due to the solenoid are now at right angles,
and the /3-particles move across them both. As they are
emitted in all directions by the radium fluoride, they cross the
magnetic field at different angles with its direction. If we
consider one which moves with velocity v at an angle a
with the magnetic field H, it is subject to a force qvH sin ale
normally to the plates of the condenser, while, at the same time,
it is subject to a force ^E due to the electric field E. These two
forces will balance one another if vH sin a = cE, and, if this
is so, the electron will move in a straight line and so be able
to pass right through the narrow gap between the plates of
the condenser. (The gap between the condenser plates being
so narrow compared with their length, it may be taken that
the electrons emerging at a given angle a have a very definite
velocity given by this equation.) On emerging they will be
subject to the magnetic field only, and will therefore be
deflected from a straight path.
According to the above theory there should be an accelera-
tion perpendicular to v and to H of magnitude ^z/H sin a/mc{3,
and the distance to be traversed is d/sin a, d being the distance
from the condenser to the plate.
Thus to a first approximation the deflection z will be
given by
z = -J(^/Hsin ajmcff) (d\v sin a)2
In this expression for z every factor is independent of a ex-
cept /3-1 which is equal (i - E2/H2 sin2a)£.
Thus the electrons will meet the plate in a curved line
with a maximum deflection for the rays incident normally,
that is, for a = |TT.
The measurements in the third part of the table below
were made by the application of this method, which has proved
capable of much more accuracy than the original method of
Kaufmann.
54. TABLE OF RESULTS.
For the purpose of comparison of results obtained by
different methods and for electrons of different velocities, it is
5*
68
RELATIVITY AND THE ELECTRON THEORY
convenient to put together the values of the ratio q\m> m being,
as above, the apparent mass for small velocities, the equations
of motion being assumed to be those obtained above by the
application of the Principle of Relativity. It will be seen that
there is a remarkable agreement between the values obtained.
The first four lines in the table give the ratio q\m for com-
paratively slow rays, as the cathode rays, and the rays emitted
by a glowing body. For these there is no question as to the
dependence of inertia upon velocity.
Next is given the value which is predicted by the best
measurements of the Zeeman effect, interpreted on the hypo-
thesis that the spectral lines are due to the vibrations of an
electron.
The last part of the table gives the values obtained from
£-rays of velocity comparable with that of light.
vjc
q\m x
10 -7
J. Classen, "Verb. d. Deut. Phys. Cres.," 10 (1908), p. 700 .
J. Malassez, " Ann. d. Chim. et de Phys.," 23 (1911), p. 231
•
*
i'773
1769
A. Bestelmeyer, "Ann. der Phys.," 35 (1911), p. 909 .
•
1766
E. Alberti, "Ann. der Phys.," 39 (1912), p. 1133 .
*
1766
Weiss and Cotton, " J. de Phys.," 6 (1907), p. 429
t
1767
P. Gmelin, "Ann. der Phys.," 28 (1909), p. 1079 .
1771
A. Bucherer, - Ann. der Phys." {£ <'$*>; £ JJ| . .
"Si
•69
i752'\
1767 /
R. Wolz, " Ann. der Phys.," 30 (1909), p. 273
{?
17676-1-
17672;
C. Schaefer (G. Neumann), " Phys. Zeit.," xiv (1913), p. 1117
•4 to -8
17676 3
In addition to the experiments of Wolz, Bucherer, and
Neumann, reference should be made to those of Hupka.4
* Negligible.
f Assumed to be negligible; experiments on Zeeman effect.
1 Bucherer, in a later paper, points out a correction which should be made
for the edge effect of the condenser. This would probably raise this result to the
same value as the others. The following experiments of Wolz took particular
note of this correction. 2 ± '0025.
3 Of 26 results obtained, all lie between 175 and i'8 ; all but four between
175 and 178. Possible error, + '0025. See Figure (8) below.
* « Ann. d. Phys.," 31 (1910), p. 169.
MECHANICS AND THE PRINCIPLE OF RELATIVITY 69
1
1
O j
/
£
(0
9
1
I
1
/
6
JO
t4-
0
1
0 JJ?
/
TO
3
r.
r
i
X
0 ^OvP
^ JZ
X ^
t
/
ill
<r°
7
i
i
I 5
2 ^
n ^
*
/
0 ^g oo
**- 2
U) ^ W
V. ft)
i
i
/
O <fc 3
^ —
CQ «
t
i
I
i
f- .
A S
JC
1
b
i
6
i
<v
>
3
o
a
§ £
0
1
*
1
r
*
J
9
in
II
70 RELATIVITY AND THE ELECTRON THEORY
This author does not give an absolute value of qjm. He
shows, however, that over a considerable range of velocities
the formula of Lorentz gives a very constant value for qjm,
e.g. in one set of Hupka's experiments in which the values of
vie varied between -346 and "501 the calculated values of q\m
did not vary by more than '03 per cent.
55. THE SIGNIFICANCE OF THIS VERIFICATION.
The experiments which have just been discussed were, in
the first instance, undertaken by Kaufmann merely with the
intention of examining how far the inertia of an electron was
due to the charge which it carried. It had been suggested by
Sir J. J. Thomson as early as 1881 that the inertia of a body
was increased by the presence of an electric charge on it. The
result of Kaufmann's first experiments on the cathode rays
was to show that, within the range of velocities there present,
the whole inertia of the particles varied exactly in the same
way as the electro-magnetic part of it should do, as calculated
by Abraham from the hypothesis that the electron was a
spherical distribution of charge.
Three years later, on the publication of the suggestion of
the contracting electron to explain the results of the Michel-
son and Morley experiment, Kaufmann repeated his ex-
periments with the /3-rays in which higher velocities were
accessible, with the object of distinguishing between the two
theories. It cannot be said that the results obtained were en-
tirely in favour of Lorentz' s hypothesis ; but more recent ex-
periments, as the figures reproduced in the above tables show,
were distinctly so. We may now say with certainty that the
contraction hypothesis is not contradicted by experiment.
But to say that the results confirm the hypothesis ' that the
whole of the inertia is attributable to purely electro-magnetic
origin ' is to forget the nature of the assumptions that are made
in making the calculations.
The argument, as given by Lorentz, rests essentially on
the hypothesis of the contracting electron. , It was remarked
above that to make this hypothesis is in reality only to shift
the difficulty of explaining the facts back from the FitzQerald
MECHANICS AND THE PRINCIPLE OF RELATIVITY 71
hypothesis, which refers to concrete matter, to the same
hypothesis about a purely conjectural element in the constitu-
tion of matter, and there it must be left If we accept the
crude picture of the electron as a spherical shell carrying a
surface layer of charge, we must not ask how this shell is held
together against the mutual repulsions of the parts of the charge
which is carried ; in fact, we hardly dare think of the charge
on the electron as divisible, much less may we ask why the
electron contracts, and whether there are ' forces ' which cause
it to do so. All that we can do is merely to assume that it is
exactly so constituted as to fall in with the experimental facts.
There is then something over and above purely electro-
magnetic theory required for the complete explanation of the
facts observed. The simple comparison of the results of
Abraham and Lorentz, which give entirely different values for
the apparent inertia of an electron whose velocity is compar-
able with that of light, shows that the choice of this additional
element to supplement the electro-magnetic laws is one which
may seriously affect the results predicted by those laws. This
is, for example, clearly shown by the lower curve in fig. 8,
which exhibits side by side the values of q\m calculated from
the experimental results of Neumann (i) by Lorentz's theory,
(ii) by Abraham's theory. The latter theory differs only from
that of Lorentz in the assumption that the electron is always
a sphere, whatever velocity it is supposed to have.
Thus the agreement of experiment with the predictions
of Lorentz's theory confirms the general correctness of his
hypothesis. But it cannot be said to confirm the particular
hypothesis of the contracting electron, which is only one of
many that are consistent with the Principle of Relativity. In
any case, it cannot be said that Lorentz's theory is a purely
electro-magnetic one.
The results recorded above are therefore to be considered
as confirming the general hypothesis of relativity as a principle
that goes beyond the mere invariant form of the electro-
magnetic equations.
CHAPTER VI.
MINKOWSKI'S FOUR-DIMENSION VECTORS.
56. INTRODUCTORY.
THE position at which our discussion has now arrived is this.
We saw in Chapter III. that, confining our attention to the
facts of the propagation of light in free space, the possible
methods of partitioning time and space were not independent
of one another, that there was no unique sense to be attached
to the phrase " simultaneity of events ". This has been
followed by a consideration of physical phenomena of a varied
nature, with the result that we have arrived at the conclusion
that the ambiguity in, and the interdependence of, the modes
of measurement of time and space has, up to the present, not
been removed.
Minkowski says of the situation so created : "From hence-
forth^ space by itself, and time by itself, are mere shadows, and
only a kind of blend of the two exists in its own right"?-
All future discussion of the exact relations of the constitu-
tion of matter must be influenced by the general evidence for
the validity of the hypothesis of relativity. If the principle is
not universal it will sooner or later prove its own destruction
by predicting results which are at variance with experiments.
We must remember, however, that the principle is on a dif-
ferent plane from such " physical " laws as those embodied in
Maxwell's equations or the law of gravitation. It is a general
] " Raum und Zeit," 1908. Reprinted in " Das Relativitats prinzip," Leipzig^
72
MINKOWSKPS FOUR-DIMENSION VECTORS 73
mode of thought. We are not concerned with any particular
quantitative law, but with the question as to whether all the
quantitative laws of physics conform to a certain common
criterion.
There is one such common criterion which is commonly
accepted. We may call it the " vectorial" criterion, or the
criterion that a mere change of orientation of a physical
system in space will not alter its intrinsic properties. Space
is said to be isotropic as far as it is merely a mode of ordering
physical properties. The result is that, although we often
employ a particular set of co-ordinate axes as a frame of re-
ference, we recognize that the form of the equations which
embody our physical laws must be one which is unaltered if
we turn our axes into any other angular position. What is
known as " vector analysis " is a notation and calculus which
recognises this principle from the beginning, and which is
adapted specially and only to the expression of such laws as
conform to it.
The criterion of relativity is very analogous to this, and
was, in fact, shown by Minkowski to be expressible as a
generalization of it. An attempt will now be made to present
the main idea introduced by Minkowski.
57. THE LORENTZ TRANSFORMATION AS A ROTATION OF
AXES.
The change in the measures of space and time, which we
have called a Lorentz transformation, may be written thus —
x = x cos 0 — u sin 6, y = y,
u = x sin 6 + u cos 0, z = #,
where, /standing as usual for *J(- i),
u = ict, u = ict', cos e = * sin 6 = ~
the last two quantities satisfying the equation
cos20 + sin20 = I.
The analogous equations in three variables,
X = x cos 0 - 2 sin 0, y — y,
z = x sin 6 + z cos 6.
74 RELATIVITY AND THE ELECTRON THEORY
are the equations which correspond to a simple rotation of the
axes of co-ordinates through an angle 6 about the axis of y.
We may then think of the Lorentz change of variables as
a rotation of the imaginary set of axes through the imaginary
angle 9 in the four-dimension space in which the co-ordinates
are (ar, y, z, icf). This is, of course, only a way of visual-
izing what are in effect algebraic processes involving only real
quantities.
58. EXTENSION OF THE IDEA OF A "VECTOR".
Again, in three dimensions we speak of * vector quantities/
as directed quantities which combine according to the parallelo-
gram law.
If a certain vector is represented by components (X, Y, Z)
along the axes of (x, y, z\ and by the components (X', Y', Z')
when the axes of (x , yt z) are used, the relations connecting
(X', Y', Z') with (X, Y, Z) are exactly the same as those con-
necting (ar, y\ z') with (x> y, z). A vector quantity can be
exactly represented by a line.
So Minkowski speaks of '4-vectorsJ as quantities with
four components which when the frame of reference is changed
are subject to exactly the same equations as are the space-
time co-ordinates (ar, y, z, u). As a particular case, for the
Lorentz transformation the 4-vector (X, Y, Z, U) is changed
to (X', Y', Z', U'), according to the equations
X' = £(X - M/U), Y' = Y, Z' = Z, U' = /3(U + wX/*8),
where the velocity v is arbitrary, and the direction of the axis
of x is also an arbitrary direction in space.
These equations are equivalent to
X' = £(X + z/T), Y' = Y, Z' = Z, T = /3(T + vX/*8),
where U = zVT, and U' = icl'.
59. EXAMPLES OF 4- VECTORS.
Let two point-instants be in a certain frame of reference (xu ya zlt /x)
(*ai y*> *a, /a)- since (*n y-a *» *#i)» (xv ?*> zv ict'b are 4-vectors, so is
their difference
* - * *<* - -
MINKOWSKPS FOUR-DIMENSION VECTORS 75
If we write for the differences (8x, Sy, 8g, zVS/), remembering the
fundamental property that
is an invariant (p. 40), it follows that
(bx, S/, dg, icdtytM
is a 4-vector ; that is, for a moving point
U = <(ux, uy, ux, ic) is a 4-vector
where < = (i - (ux* + u* + u»*)l*}-*.
More generally,
if I = (X, Y, Z, U) is any 4-vector
then j = * §. (X, Y, Z, U) is a 4-vector.
where X, Y, Z, U are supposed to be any functions of the space-time co-
ordinates (*, y, z, /) of a certain moving point.
The following may be stated to be 4-vectors without proof : —
e = {e + [uh]/f, i(ue)/^}
D = (h - [ue]/<r, i(uh)/*},
where the symbols have the same meanings as in the fundamental equa-
tions, and the first three components of the 4-vector are the components of
the ordinary vectors specified.
60. SPACE AND TIME AS Two ASPECTS OF A UNITY.
We now see what Minkowski had in mind when he spoke
of " only a blend of space and time existing in its own right ".
The 4-vector is thought of as a single entity, just as ordinarily
we think of a force as a single entity. The components of a
force do not exist by themselves. They are only convenient
means of specifying the force which is one and definite. Just
in the same way space and time co-ordinates are to Minkowski
only particular and complementary aspects of a single fact or
occurrence.
Analytically Minkowski transports himself to a space of
four dimensions in which the distinction between space and time
vanishes. In this four-dimensional region, the whole of space
and time is portrayed in one construct. The motion of a moving
point through all time is represented by a single curve, the
points on the curve being ordered to correspond with the suc-
cession of events in time, but the interpretation of the curve
as representing an ordinary motion is not unique; it depends
upon the choice of the direction in the four-dimensional region
which is chosen to be the time axis.
76 RELATIVITY AND THE ELECTRON THEORY
Thus three-dimensional kinematics becomes four-dimen-
sional geometry. This relation extends further, it reaches into
the domain of mechanical quantities. Three-dimensional dyna-
mics can be interpreted as a four-dimensional statics, an
exact generalization of three-dimensional statics. An example
of this will be given shortly.
61. A SECOND KIND OF FOUR-DIMENSIONAL VECTOR.
Before passing on to this, however, it will be well in passing
to indicate the way in which Minkowski, thinking in terms
of this four-dimension space, pictures the significance of the
relativity of the electro-magnetic vectors e and h.
In ordinary three-dimension space an element of a straight
line — i.e. a one-dimensional region — and an element of area
— i.e. a two-dimensional region — may be equally well repre-
sented as vectors. But in four dimensions, of the elements
of regions of one, two, and three dimensions, only the
first and third are capable of representation as what we have
called 4-vectors. A two-dimensional element will have to
be represented by six components, corresponding to the
number of ways of choosing two out of the four co-ordinates
of a point.
Now Minkowski was able to show that the transformations
(B, p. 48), affecting (exy ey, ez~) (ikx, ihy, ihz} are exactly of
the same form as those which affect the six components of a
two-dimensional region in four-dimension space when the
fundamental geometrical transformation of (ar, j, z, /) is carried
out Hence he introduces a new conception ; he defines a
1 6- vector' to be a quantity with six components which trans-
forms in exactly this way. Then the validity of the trans-
formation (B) is contained in the phrase
3f =* (e, zh) is a 6 -vector.
One property which follows from this, and which may be
deduced immediately from (B), is that
jf 2 = e2 - h2 is invariant.
We might here extend the remark of Minkowski quoted
above (p. 72), and say that "from this point the electric intensity
and the magnetic intensity apart from each other have no signi-
MINKOWSKPS FOUR-DIMENSION VECTORS 77
ficance, but that only a single quantity compounded of the two
exists in its own ru
62. THE PRINCIPLE OF LEAST ACTION A GENERALIZATION
OF THAT OF LEAST POTENTIAL ENERGY.
It is now possible to give a striking example of the way
in which these conceptions give to dynamical equations a
symmetrical form which is an exact generalization of the
ordinary vectorial form of statical equations.
In electrostatics the ' total energy ' of a field may be written
where dV is an element of volume1 ; and there is a well-known
theorem that the electro-static field is such as to render this
total energy a minimum subject only to the restrictions imposed
on it by a given distribution of charge. The generalization of
this is : —
If d\& represent an element of the four-dimensional space, and
JjfVU) be called the 'total action] then the electro-dynamic field
is such as to make this total action a minimum, subject only to
the restrictions implied by a given distribution of moving charges.
This is only the expression in the language of four vectors
of the known theorem that the equations of the electro-magnetic
field are such as to make the time integral of the difference of
the magnetic and electric energies a minimum. In ordinary
notation this integral is
In comparing these theorems we may note that whereas in
the former the ' total energy ' is a quantity which is indepen-
dent of the frame of reference, so in the second the 'total
action ' is a quantity that does not change when we alter the
space-time system of measurement.3 For it has been pointed
out above that 3f2 is invariant, and the element of integration
d\D is unaltered by a turning round of the axes.
1 Remembering that we are using Lorentz's units.
2 See Larmor, «« ^Ether and Matter," pp. 82 if.
3 In changing from the variables (x, y, z, t) to (#', y', z', t') we substitute for
<W the quantity ^*] •?; *; *) dW, and it is easily seen that ^|*' •?' *' *| is
unity.
78 RELATIVITY AND THE ELECTRON THEORY
63. THE NEW MECHANICS— c FORCE' AND 'WORK' AS
Two ASPECTS OF A SINGLE CONCEPT.
In the classical mechanics there have always been two
strains of thought. The two aspects of * force ' as ' the time
rate of change of momentum,' and as ' the space rate of change
of energy,' have with different writers been given different
degrees of prominence. Galileo developed the former, Huy-
ghens the latter. In the light of four-dimensional vectors
the two ideas become unified, and differ only as partial aspects
of a greater concept, just as space and time are unified and
become partial aspects of the whole extension of the universe.
The equations of motion for a body moving as a whole
with velocity u without rotation are
4-.k
di - k>
g being the momentum, and k the moving force.
The energy equation is - = (ku)
at
w being the energy, and u the velocity.
In order to bring these equations into line with the Prin-
ciple of Relativity, we have only to assume
(i) that if g is the momentum and w the energy then (g, iwjc)
is a ^-vector, Q ;
(ii) that if k is the force acting on the particle and u its
velocity then
(k, * (ku)/£) is a ^-vector, ft ;
(iii) that the ordinary equations of Newton are only an
approximation to the more exact equations
We have seen that (&r, By, 82, ic&f) being a 4-vector, Stlic
is an invariant.
Hence if f is any 4-vector, and Sf is a small change in it,
K -5- or, in the limit, K —
dt dt
is another 4-vector.
Now, if (ft) = (k, z(ku)/£) be a 4-vector, the equation
= K~dt
MINKOWSKPS FOUR-DIMENSION VECTORS 79
is an equation of invariant form, that is, it expresses relations
which are in accord with the Principle of Relativity.
The first three components of this equation give
*-«*?
dt
which if (vie)2 be neglected reduces to the Newtonian form,
while the fourth component is, to the same order,
/i \ dw
(ku) ' -di
which is the usual equation of energy.
The distinction between the schools of Galileo and Huy-
ghens, between 'force* and 'work/ is here lost, and we see
them each as partial aspects of a greater whole.
64. RELATION BETWEEN MOMENTUM AND ENERGY.
The stipulation that Q shall be a 4-vector requires us to
modify the usual relations which connect the momentum and
energy of a moving particle with its velocity. Let us make
the assumption that for a particle at rest the momentum is zero.
Then for that case
QO = (o, o, o, iw0lc]
where w0 is the energy of the stationary particle.
Now, for any particle whatever we may choose a frame
of reference of which the origin has momentarily the same
velocity as the particle. The extended velocity vector
U = /e(u, ic) then becomes
Uo = (o, o, o, ic\
since in this frame of reference the particle is at rest.
The two vectors, Q0, U0 are thus in the same direction, and
we may write
Qo=~Uo.
Now the equality of two vectors is an invariant relation ;
so that, whatever the frame of reference, we must have for the
same particle
8o RELATIVITY AND THE ELECTRON THEORY
The last equation resolves into
*^ = ^—
and w = KW. W>
(I - u
g and z^ being connected by the relation
g = wu/c*.
Of these results we may note the following consequences
if we neglect higher powers of u/c than the second : —
(i) w = w0(l + iu2/<^) = w0 + ^mu2,
(ii) g = w0ult? = muy where m = wjc?.
The quantity m here enters in exactly the same way as
the Newtonian mass ; thus we are led to contemplate a relation
between the mass of a particle and its energy. That it is a
possible relation appears from the following figures. If the
particle were to give out energy, as in the case of radioactive
bodies, the apparent mass should, according to this relation,
diminish. But if we take the actual rate of loss of energy by
a gram-atom (225 grms.) of radium in its disintegration,
which is about io12 ergs per hour, and divide by <?, the
rate of diminution of the mass would be only about io~5 gr.
per year, which is quite inappreciable by ordinary methods.
In the case of large velocities the concept of ' mass ' be-
comes meaningless, but we still have the relation
g = wu/<?
which indicates that the convection of energy by a moving
body implies an amount of momentum equal to the product
of energy and velocity divided by &.
Thus if a body moving uniformly is radiating energy, it is
also losing momentum. The equation
then indicates that there is a force acting on the body even
though its velocity remains constant.
It has sometimes been suggested as a parodox, that whereas
ordinary electro-magnetic theory clearly indicates a resistance
MINKO WSKPS FO UK-DIMENSION VECTORS 8 1
to the motion of a radiating body through the aether,1 yet the
Principle of Relativity must allow that a body radiating equally
in all directions must be capable of remaining in uniform
motion through the aether if subject to no external force.
The above analysis resolves the paradox ; it admits the
resistance, but finds it to be equal to the rate at which mo-
mentum is lost owing to the radiation of energy, even though
the velocity be unchanged.
This proposition of the momentum or inertia of energy is
one which, in the development of mechanics subject to the
Principle of Relativity, is universal. A familar instance is to
be found in the mechanics of the free aether as ordinarily
stated. Poynting, in an analysis of electro-magnetic energy,
suggested that, at any point of free space, there is a >flux of
energy represented by the vector £[EH] ; while later, Abraham
showed that we may maintain the proposition of the conserva-
tion of momentum as between aether and electric charge if we
allow of .a distribution of momentum in the aether of intensity
[EH]/£ Here again the flux of energy is equal to the mo-
mentum multiplied by ^2, in exact agreement with the cor-
responding result for a material particle.
65. THE TROUTON-NOBLE EXPERIMENT.2
There is an apparent contradiction, somewhat similar to
that referred to in § 64, in the case of a radiating body in uniform
motion, which arises in respect of the moment of the forces
acting on a system in uniform motion. It can be shown
that, in order to reconcile the mechanical principles of
conservation of momentum and conservation of energy in
the aether with the known laws of propagation of electro-
magnetic disturbances, it is necessary to assign to the aether
a velocity which is not in general in the direction of the
momentum. In the same way it is possible so to modify
mechanical equations of any deformable continuous system,
using the same methods that have been indicated above for
modifying the dynamics of a single particle, that they may be
in accord with the Principle of Relativity.
1 See Larmor, " International Congress of Mathematicians," Cambridge,
1912, p. 216. 2 See p. 58.
6
RELATIVITY AND THE ELECTRON THEORY
In doing this two important results emerge. The first
is that the relation referred to above between the flux of
energy1 and the momentum must be made universal. The
second is that, in general, the momentum per unit volume at
any point of the medium need not be in the direction of the
velocity of the medium at that point.
In a case where this difference of direction arises it is easy
to see that, for an element of volume moving with uniform
velocity and with constant linear momentum, the moment of
momentum about a fixed point is not constant as it is when
the directions are the same. For in the figure let P, P' be
the positions of the element at two consecutive instants, so
that PP' = vSt, and let the linear
momentum be in the direction NP
and of constant magnitude g.
Then it is clear from the figure
that in the time 8t the moment of
the momentum about O diminishes
by g x NN', that is, by gv sin 6&t,
a quantity which only vanishes if g
and v are in the same straight line.
Thus in the mechanics of the
Principle of Relativity we shall not expect that the moment of
momentum of a body moving uniformly relative to a given
frame of reference will always remain constant. Exactly such
a case arises in the conditions of the experiment of Trouton
and Noble (see p. 58). Here a charged plane condenser is sup-
posed to be placed obliquely to its direction of motion relative
to a given frame of reference. The classical theory shows
clearly that there must be a couple acting on it in a sense
which would, according to ordinary dynamics, tend to turn it
so as to be normal to the direction of motion. But if the
dynamics of the Principle of Relativity are applied to this
case we find that the couple is exactly equal to the rate of
change of the moment of momentum which arises as above
from the fact that the linear momentum, though constant, is
in a different direction from the velocity. Thus the mechanical
1 Flux of Energy = Momentum x c3.
MINKOWSKPS FOUR-DIMENSION VECTORS 83
conditions are completely fulfilled without any rotation of the
condenser being set up.1
66. EXTENSION OF ANALOGY BETWEEN DYNAMICS
AND STATICS.
In § 62 we have shown that the relation of the laws of the
electro-dynamic field to the principle of least action can be
looked upon as the extension to Minkowski's four-dimensional
world of the relation of the laws of the electro-static field to
the principle of least potential energy. In that statement,
however, no account was taken of the mechanical constraints
imposed upon the charge or current ; the distribution of these
was assumed to be given, and the variation of energy or
current was subject to the given distribution.
In the electro-static case, if the charges or conductors
carrying them are supposed to be moved, we know that the
increment in the electro-static energy is equal to the virtual
work of the non-electrical forces which maintain the equili-
brium of the conductors against the electrical attractions and
repulsions. Or again, if the non-electrical forces are con-
servative so that we may speak of a corresponding potential
energy, the variation of the total potential energy in a small
displacement from the equilibrium position is zero. We note,
of course, that the total energy is a scalar quantity, and is
independent of the choice of any particular system of reference.
In the dynamical case we have the corresponding theorem
that if A! represents the electro-dynamic action as defined
in § 62, and if there be a non-electrical action A2, arising from
non-electrical inertia and forces, then if A = Ax + A2 is called
the 'total action,' then, for any slight variation of the currents
and motion of the carriers of charge and current from the actual
motion which takes place, the variation in A is zero.
This very general proposition must, as in the simpler case,
be independent of the choice of a frame of reference ; that is,
if we maintain the relativity of all the phenomena, it must
have an invariant form for any change of axes in Minkowski's
four-dimensional space. Now we know, as in § 62, that Alt
1 For a fuller discussion see the author's larger work (Camb. Univ. Press,
1914, p. 170).
84
RELATIVITY AND THE ELECTRON THEORY
the electro-dynamic part of the action, is an invariant Hence
the Principle of Relativity requires that if there is a non-electri-
cal action A2, it too must be an invariant.
70. One simple development may be set down as an indication of the
application of the above result.
If k = (kx, ky, kz) is a typical force acting at a point (x, y, z] at time
/, and we consider a slightly varied motion of the system in which the co-
ordinates of the same point are (x + &r, y + ty, z + dz) at time (t + S/),
then the variation in the action arising from k is known to be
where u is the velocity of the point in question (see e.g. Routh, " Rigid
Dynamics," Vol. II, p. 280).
In Minkowski's notation this can be written ftSr. where ft = (k, *'(ku)/c)
and SC = (&r, 5y, 8z, icdf). The invariance of this quantity, St being an
arbitrary 4-vector, requires that ft shall also be a 4-vector.
APPENDIX I. — THE FUNDAMENTAL EQUATIONS IN
SYMMETRICAL FORM.
It may be worth while to set down the equations of the field as Min-
kowski arranges them. For fuller details see the author's larger work,
Chapter IX.
Writing
(F,,, Fzx, Fxy, Fxu, FyU)
= ( - H,, -
- Fsyt etc. ;
- H,,
and F
also (S^, Sj,, S2, SM) = p (ux, uy) uz, ic}\c,
and u = icf,
the equations of the field become
c)Fzx
~W
+
3F,
du
9*
Or
'du
~3T"^r
= o
= o
MINKOWSKPS FOUR-DIMENSION VECTORS 85
The symmetrical form of the equations is now obvious. They may
be easily shown to have an invariant form, and may be looked upon as
the generalization of the vectorial equations of the electro-static field
dive = p
curl e = o.
(Sx, Sy, S*, SM) is easily seen to be a 4-vector by means of the equations
(a) and (C), p. 48.
PART II.
THE GENERAL PRINCIPLE OF
RELATIVITY
CHAPTER VII.
THE GENERAL THEORY.
67. THE NEED FOR MORE RADICAL RE-CONSTRUCTION.
WE are now in a position to appreciate the way in which
a number of different difficulties united in calling for the
comprehensive reconstruction effected by Einstein under the
name of the General Principle of Relativity. The Special
Principle was unsatisfying in two ways.
On the one hand, while demanding a drastic revision of
our ordinary ways of thinking about space and time, it was
by no means thorough-going enough for the philosopher.
The old problem of absolute acceleration was left exactly
at the same point at which Newton had left it
On the other hand, the new principle had not been able
to go any further than the electron theory of matter, of
which it was a natural complement, in interpreting the
nature of gravitation.1 The electro-magnetic theory seemed
fairly well able to account for those properties in which
various kinds of matter differ from one another, as, for
example, thermal and electrical conductivity, optical dis-
persion, magnetizability, as well as being a clear basis of
chemical relations. But of the two universal and common
properties of matter, inertia and gravitation, the latter remained
obstinately aloof. At the same time, however, the growing
conviction that the Newtonian conception of mass as an
1 For attempts to solve the problem of gravitation on the ground of the
Special Principle of Relativity, see Poincare, " Rend, del Circ. Mat. di Pal.," 21
1906), p. 166; also de Sitter, "Monthly Notices of Roy Astr. Soc.," March,
1911, p. 388 ; also various papers by Nordstrom, and others, in " Phys. Zeitsch.".
86
THE GENERAL THEORY OF RELATIVITY 87
invariable measure of the inertia of a body was at best an
approximation, began to raise the question as to whether the
gravitational attraction between two bodies would not also in
some way depend upon the frame of reference adopted. For
the new theory, supported by the experiments on the mass of
the electron (Chapter V.), showed that the mass of a body for
the purposes of inertial phenomena was certainly a function of
the velocity, and probably of the internal energy of the body.
The question at once arose : Is the gravitational mass of a
body, the quantity which measures the power of one body to
attract another, also a variable quantity ? If so, is it always
proportional to the inertia-coefficient ?
These two questions bear one upon the other at various
points. For instance, if Newtonian dynamics is only approxi-
mately correct, then the partial relativity of dynamical relations
is only approximate too. Further, if energy of all kinds is
inseparable from momentum, if light has inertia, may it too be
subject to gravitation ? This would imply a possible deviation
from the electro-magnetic equations "of Maxwell out of which
the early Principle of Relativity arose. That being so, the
velocity of light might not in strictness conform to the hypo-
thesis of a fixed and uniform velocity.
This, of course, would be expected by one who adopted
the point of view that space and time are merely arbitrary
modes of ordering phenomena, and that from an a priori
point of view we can know as little about absolute rotation
or acceleration as about absolute velocity.
But once concede that the velocity of light is not an ulti-
mate, universal, and invariable quantity, and the whole basis
of the Special Principle of Relativity needs revision. For the
Lorentz-Einstein transformations were just selected as that
group which preserved this one metric property of the universe
intact. Minkowski's four dimensional vectors also depend for
their validity on this special hypothesis.
68. EOTVOS' EXPERIMENTS.
The first point to be considered was that of the relation
between the two universal properties of matter, inertia, and
gravitational attraction. In the first instance, the question
88 RELATIVITY AND THE ELECTRON THEORY
asked was merely this : Is the inertia mass strictly the same as,
or proportional to, the gravitation mass ? Or more precisely,
within what limit of accuracy can we definitely affirm on
experimental grounds, that they are so ?
Here the experiments of Eotvos have been quoted,1 which
purport to show that the ratio of the gravitational attraction of
the earth upon different bodies to their inertia masses certainly
does not vary by as much as the fraction 5 x io"8; this in a
comparison between specimens of such different bodies as
brass, glass, antimonite, and cork ; bringing a mass of air into
the comparison also, io~5 is the limit of accuracy within which
it is certain that there is no variation.
The principle of the experiments is as follows : —
Two masses of different substances are fastened to the
ends of the beam of a delicate torsion balance. The beam is
adjusted to hang in an east to west direction. It hangs in
equilibrium under the attraction of the two gravitational
attractions, and the centrifugal forces arising from the earth's
rotation. The latter are proportional to the dynamical masses
of the bodies, the former to what we have called their
gravitation masses. If the dynamical masses and gravitational
masses are equal, the resultant forces (or apparent weights of
the masses) will be parallel, and so their resultant will be a
single force, which, of course, must just be balanced by the
tension in the suspending thread. If, however, the masses are
not exactly proportional, the apparent weights will not be
parallel, and so can only be combined into a single force
together with a couple about its line of action, balanced
respectively by the tension and the torsion in the thread.
The magnitude of this torsional couple will be approximately
wlfa where w is the weight of either mass, / its distance
from the point of attachment of the thread, and $ the angle
between the apparent weights — an angle which will depend
upon the inequality of the mass ratios.
Eotvos thus describes his experiment : —
" I attached separate bodies of about 30 grms. weight to
1M Mathematische und Naturwissenschaft liche Berichte aus Ungarn," Bd. 8
(1891), p. 64.
THE GENERAL THEORY OF RELATIVITY 89
the ends of a balance beam about 25 to 30 cms. long, suspended
by a thin platinum wire in my torsion balance. After the beam
had been placed in a position perpendicular to the meridian, I
determined its position exactly by means of two mirrors, one
fixed to it, and one fastened to the case of the instrument
I then turned the instrument, together with the case, through
1 80° so that the body which was originally at the east end
of the beam now arrived at the west end. I then determined
the position of the beam again relative to the instrument."
If a couple such as has been mentioned above existed, it
would on turning the instrument round act in the reverse
direction. Here Eotvos continues — "If the resultant weights
of the bodies attached on either side pointed in different
directions, a torsion in the suspending wire should ensue.
But this did not occur in the case in which a brass sphere was
constantly attached to the one side, and glass, cork or crystal
antimony was attached to the other, and yet a deviation of
i-6o,oooth of a second in the direction of the gravitational
force would have produced a torsion of one minute, and this
would have been accurately observed."
The actual deviation of the pendulum from the true vertical
due to centrifugal force is about 360 seconds, so that to the
accuracy of I in 2 x io7 the ratio of dynamical to gravitational
mass is shown to be the same for the two bodies. .
More recent experiments by Zeeman1 on bodies with radio-
active properties show the same result to a degree of accuracy
varying between 2x1 o~7 and 3x1 o~8.
69. GRAVITATION AND A BEAM OF LIGHT.
If we are led by these considerations to think that the
internal energy of a body must be as much an essential factor
in determining its susceptibility to gravitational influence as it
is in determining its inertia, a natural sequel is to suppose that
the purely non-material energy in a light wave, and the inertia
which it has been shown to possess, will also in some way be
subject to gravitation also. But a difficulty at once presents
itself. In the case of material bodies, gravitational action
always shows itself in accelerated motion. It has been pointed
Zeeman, " Verb. d. k. Ak. van Wet," 26 (1917), p. 451.
90 RELATIVITY AND THE ELECTRON THEORY
out (§ 67) however that, as one of the consequences of the special
Principle of Relativity, we are no longer bound to think of the
momentum of a body as being in the same direction as, or
bearing a constant ratio to, its velocity. The velocity may
remain constant, but a change in the internal energy or stress
may yet produce a change in the momentum.
Thus we have to ask the question : If there is a gravita-
tional action upon light, will it be evidenced in a change of
velocity, or simply in a change in the nature of the disturbance
which is being propagated ?
To ask the question is to admit that either alternative may
prove true. It is in following the consequences of the first
possibility that we see the generalization of the Principle of
Relativity open out.
The first step in the quest is to consider the consequences in
the ordinary material phenomenon of weight. We come back
to Galileo and his discovery that falling bodies have all the
same acceleration. Does light then share in this acceleration ?
Another way of putting the same question is this : —
" To an observer who was falling freely in a gravitational
field, would a ray of light appear to be travelling uniformly in
a straight line?"
This form of the question suggests a further and more
general consideration. Would it be true that to such an observer
all perceptible influence of gravitation on physical phenomena
of every kind would be lost ? Or, if the acceleration were only
a part of that of free fall, would the gravitational influence
on all kinds of phenomena be diminished in the same ratio ?
If that were the case it would appear that the nature of
gravitational influence is strictly comparable with the nature of
those effects that arise from an acceleration of material systems.
Gravitational forces would be of the same nature as the so-
called centrifugal force apparent in the case of rotating
systems, but dismissed by the Newtonian as fictitious.
70. ECONOMY OF THOUGHT.
The whole of the discussion of the special Principle of
Relativity was guided by the main thought of preserving the
THE GENERAL THEORY OF RELATIVITY 91
simplicity of the relations conceived to hold between the
physical phenomena. No one ever denied that other frames
of reference with any accelerations or rotations and arbitrary
changes of co-ordinates might be adopted at pleasure by those
who did not mind giving up that simplicity. But once it is
granted that those simple relations cannot in all circumstances
be maintained as strictly true, the case for a limitation of the
allowable frames of reference breaks down. As long as
Newtonian dynamics were the last word in physical theory,
it was of no avail for the philosopher to say that the mind
could not conceive of absolute rotation. The scientist could
only reply that nature certainly made a distinction between
one frame of reference and another ; that for some frames
Newton's simple relations did hold, and for others they did
not. But now the retort is invalid. The Newtonian equations
are only approximately true in any case. The apparent
simplicity arises because the frame of reference is chosen in
such a relation to the phenomena in question that the re-
maining terms become inappreciable.
71. A FRESH START.
With this in mind we must turn back and see how far pro-
gress is possible along the lines of supposing that all precon-
ceptions are given up as to the precise form of physical laws.
We are to remember that the use of space and time co-
ordinates is simply a means to describing the relations that
hold between physical phenomena ; and that, as was said
above (§§ 26 and 27), if we assume in advance a specific form
for some physical law, we are in reality as much describing
the relation of our system of measurement to the phenomena
as describing the sequence of phenomena themselves. This
will perhaps be made a little clearer by an illustration from
two dimensions.
Two maps of a given part of the earth's surface, say a map
on the Mercator projection, and a map on the stereographic,
can easily be recognized, in spite of their difference in shape,
as representing the same region. The ordering of places is
distorted but not broken. If, on the map, the curves are
92 RELATIVITY AND THE ELECTRON THEORY
drawn which represent the shortest distance between given
points (the great circles), the form of these curves will enable
us to recognize what system of projection is being used. In
other words the map embodies within it not only some in-
formation about the reality which it pictures ; but, if the
geodesies are drawn upon it, it gives also information about
the mode of representation.
We are to think of the metrical description of physical
phenomena as of a four-dimensional map of the history of the
universe. The ' world lines/ as Einstein calls them, are the
curves which in the map represent the history of identifiable
particles. If we do not make any assumptions in advance as
to the form of the world lines and their mutual relations, our
mode of representation is quite arbitrary. By changing to
another mode of representation the world lines may take quite
, different forms. The hypothesis of relativity assumes that
no one form is more true or actual than another. They are
all equally valid representations of reality.
It seems at first sight that, if we contemplate the distortion
and stretching of the map to an arbitrary degree, there is no
possibility of any metrical property of the universe being
equally well represented on the map whatever form it may
take. Yet an exact physical law must be a statement of
equality which holds whatever method of expressing it is em-
ployed. The problem set to himself by Einstein and satis-
factorily answered with the aid of pure mathematics already
devised in other connections, was just that of finding mathe-
matical relations which satisfied this criterion.
72. AN ESSENTIAL CONDITION OF MATHEMATICAL
REPRESENTATION.
We must suppose that there is at least this much in
common between all the possible pictures that may be formed
of the universe and its history, that there is what we may call
an exact correspondence of events. If something happens, a
collision of two atoms, a tick of a clock, each observer will
mark it as happening at a certain place and a certain time ;
and though they do not use the same methods of describing
THE GENERAL THEORY OF RELATIVITY 93
the place and the instant, there will be in each of their several
four-dimensional maps of all the happenings a definite point
corresponding to and recording this event.
It must be a definite principle that there is what mathe-
maticians call a one-to-one point-correspondence between all
possible representations of physical phenomena.
An important application of this is in respect of what has
been said above about the nature of experimental observations.
To use again the example of the reading of a galvanometer,
we merely observe a coincidence of a spot of light with a
mark on a scale. This is what we have just called an event.
In each several map of the universe the event is uniquely
recorded as a single coincidence.
Suppose that (xv x^ x^ x^ are the four variables which are
used by one observer to define particular events. To any
particular event corresponds a unique quartet of numbers as
the corresponding values of these quantities, and vice versa
to any quartet of numbers corresponds only one event. If
another observer defines the event by another set of variables
(.*•/, x^ x^ x^) the same is true of these.
Thus to a given set of values (x^ x^ x^ x^ corresponds a
unique set (x^ . . . x^) and vice versa.
73. THE COMPLETE RELATIVITY OF CO-ORDINATE
SYSTEMS.
The special Principle of Relativity has taught us there is ho
distinction in character between the time and space co-ordinates.
On the other hand, the metaphysician who has denied the
validity of an absolute scale of time, or of space, has usually
done it merely on the ground that, so long as the ordering in
time and the ordering in space are not broken, a non-uniform
stretching of the time variable or an arbitrary distortion in
space may be contemplated.
Put mathematically the metaphysician (previous to the
Special Principle of Relativity) was inclined to allow that-
taking the suffix 4 to refer to the time variable we might
allow that what one man calls a fixed point another may
take to be moving in any manner he pleases — so that
94 RELATIVITY AND THE ELECTRON THEORY
x\ ~ fi (*i> Xtb xv x±)
fi> fv fz being any functions whatever, while a non-uniform
stretching of the time scale would be given by x± = f4 (x^).
With the assimilation of space and time variables proposed
by Minkowski, the only general method of procedure seems to
be to start by allowing x^ x^ x^, x± each to be any arbitrary
function of (xly x^ xz, x±).
We return then to the point at which we started (Chapter
III, § 26) to construct all possible systems of space and time
measurement, leaving aside the hypothesis of constant velocity,
the only limiting hypothesis there employed.
74. THE GENERAL QUADRATIC FORM.
That hypothesis was bound up with the invariant form of
the quadratic differential expression
dx* + dy* + dz* - c*dt\
\ix is any function of x, y, zy t, say/(;tr, y, zt t]
Thus on substituting expressions such as this for dx ', dy',
\ dt we find that
is identically equal to an expression of the form
in which the coefficients ars are functions of x, y, 2, t. The
aggregate of these coefficients we shall call the tensor ars
(see below § 79). With an arbitrary change of co-ordinates, the
numerical values of the coefficients are not kept unaltered as
they are under the Lorentz transformations ; though the general
form of the expression as a homogeneous quadratic differential
form remains.
75. SOME CONSEQUENCES OF A GENERAL CHANGE OF
CO-ORDINATES.
Suppose now that x ', y'} z' , / are ordinary space-time co-
ordinates for the representation of physical phenomena in a
field in which the laws are of the classical form, in which, for
THE GENERAL THEORY OF RELATIVITY 95
example, a wave of light is propagated uniformly with
velocity c\ and in which there is no gravitation. In this
picture, if we follow a light disturbance,
dx* + dy"1 + dz"* - (?dt;* = o.
If the same phenomena are pictured by means of relations
between xyyt z, and t, the same disturbance will be propagated
according to the law
a^cbP + 2a^dxdy . . . + a^df = o.
If we seek for the velocity of propagation in a direction
given by
dx = Ids, dy = mds, dz = nds,
ds being the actual distance travelled we have
which is a quadratic equation for the velocity dsjdt. For the
moment the only interest in this equation is that the circum-
stances of the propagation depend upon the whole set of co-
efficients
In other words, the physical properties of space as con-
ceived by the observer using the co-ordinates x^ y> z, t will
depend upon this set of coefficients, though to the observer
using the co-ordinates x\ y, z > /, space is homogenous and
isotropic.
Further, if this latter observer notes a particle moving
freely, that is, in terms of Newtonian laws, moving uniformly
in a straight line, the observer who pictures its motion by
means of x, y, z) t will picture it as moving neither uni-
formly nor in a straight line.
For example, if
x = x cos a)/ + y sin at z = z
and y = x sin o>t - y cos wt, tf = /,
a point at rest as mapped in the co-ordinates (xt y, z, t)
will be describing a circle as mapped in the co-ordinates
(JT , /, z, /), the angular velocity being o>. Similarly a particle
96 RELATIVITY AND THE ELECTRON THEORY
moving uniformly in a straight line in the former co-ordinates
would appear to be moving in a spiral round the origin in
the latter.
Newton observing these phenomena would infer the exist-
ence of some influence exerting a ' centripetal force ' on the
moving particle. Einstein is aware of the fact that he is only
observing the facts from a special point of view, and finds that
by adopting another picture of the phenomena the necessity
for postulating the force does not arise.
More generally, taking any arbitrary transformation of
co-ordinates, and using quasi-geometrical language, the history
of a moving particle is recorded by a curve drawn in the four-
dimensional map of phenomena. By an arbitrary change of
co-ordinates the map is stretched and distorted in an arbitrary
manner, subject only to its not being split — otherwise one
event in one representation would be split into two distinct
events in the other. Newton, taking only one representation,
infers from the straightness or otherwise of the history-curves
(world lines) of particles their freedom to or subjection to
external force. To Einstein ' force ' is a purely relative term,
relative, that is, to the particular picture he employs.
But we note here, in the passage from one picture to the
other, that the same quantities which affect the unequal
velocity of propagation of light affect the motion of the free
particle. In fact, in the field of x\ y, z' , /, the velocity of
light is given by ds = o and the path of a free particle is such
that \ds is a minimum.
As pictured in the map with (x, y, 2, t) co-ordinates the
principle of exact correspondence shows that since numerically
ds = ds the velocity of light is determined by
ds = o
and the path of a free particle by \ds taking a stationary value.
Since ds* = a^cb? + . . . the coefficients alt . . . determine
the course of both phenomena.
76. THE UNIVERSAL SIGNIFICANCE OF THE TENSOR ars.
We now see why the tensor ars is associated by Einstein
with the gravitational field. Gravitation is the name we have
THE GENERAL THEORY OF RELATIVITY 97
given to the unseen influence which is evidenced by particles,
otherwise free, exhibiting non-uniform motion. Granting the
hypothesis of complete relativity we cannot say of a deviation
or change of speed in the motion of a particle, whether it is
due to the particular frame of reference used, or whether it
arises from entirely external causes. We cannot distinguish
between a * real ' gravitational field and a ' simulated ' field. We
can only deal with the actual motions perceived, and class the
whole set of quantities characterizing those motions as defining
the gravitational field, remembering, however, that the field is
relative to the observer's particular system of measurement.
In the case of what we may call a completely simulated
field, such as has been considered in the preceding section, we
see that the whole set of quantities ars is necessary to describe
the field as apparent to any particular observer. We have
seen that in this case there is a quadratic expression l
which for a given pair of adjacent events has the same value for
all observers. We may say that the quantities ars define the ex-
tent to which the space-time system used departs from isotropy
and homogeneity in regard to its relation to physical events.
In the more general case of the actual universe it is difficult
to imagine that the whole of what we have been in the habit
of calling " gravitation " can be classed as a fictitious field of
force. Nevertheless, if we are to grant the complete relativity
of our space-time notions we must grant that we are not in a
position to say to what extent it is fictitious in any given
frame of measurement. The only plan of proceeding, there-
fore, seems to be to develop a scheme of relations more general
than those which have been taken to hold in a purely fictitious
field of force, but including those as a particular case.
All that has been assumed so far of that particular case is
that when the fictitious field has been transformed right away
the quadratic expression
+ dy"1 + dz"* - <?dt'*
1 It is a great convenience, except where expressly stated, to imply that
where a suffix occurs twice in an expression a summation is to be effected over
the values i, 2, 3, 4 of that suffix.
7
98 RELATIVITY AND THE ELECTRON THEORY
has a special physical significance, in relation to the propaga-
tion of light and the motion of a free particle. In this case all
that has been said in the earlier part of this book would hold.
But if we are now dealing with the case of a field that cannot
be so transformed away, all that is left of our hypotheses is
that there may be an invariant quadratic form
£Vs rdx,
(we introduce here the letter gfs instead of ars in conformity
with the customary notation) — in which the coefficients grs are
functions of the co-ordinates, and depend upon the particular
set of co-ordinates used This expression has a value in-
dependent of the particular set of co-ordinates, and to that
extent is capable of representing some metrical property of the
physical universe which is independent of any particular mode
of observation and method of measurement. But a priori
nothing is to be assumed as to any limitation upon the co-
efficients grs. These coefficients are characteristic of the
particular mode of measurement adopted. The important
thing to notice here is that inasmuch as grs characterizes the
system of reference from which the whole universe is regarded,
there will be no class of phenomena which will not be dependent
upon this tensor. In other words, we have here a basis
from which we shall be able to consider the influence of
the gravitational field, not only upon the motions of particles,
but upon the propagation of light, the motions of electrons,
or of any other property of matter we may select.
77. EINSTEIN'S PROBLEM.
We have so far proceeded mainly by way of giving up
definite results which classical theory had regarded as funda-
mental.
It would seem that such a thoroughgoing jettison of all
definite hypotheses would leave little hope of arriving at
anything in the way of a precise physical law. Let us note
therefore, in a very much simpler instance, how what appears
to be a negative hypothesis actually suggests a known law.
Returning to Newtonian conceptions, let us think of a scalar
gravitational potential <£ with a definite value at each point
of space, and ask ourselves what possible connections may
THE GENERAL THEORY OF RELATIVITY 99
exist between such a potential and the density of matter p
also defined for each point of space — on the single hypothesis
that space is isotropic, that is, that all directions at a point are
alike ; or, in other words, on the hypothesis that the form of
all physical laws remains the same no matter how the co-
ordinate system is changed provided only that dx*+dy*+dz*
is invariant.
The problem is thus : we are given two quantities p and (f>
at every point; what types of relation between these two
quantities could possibly hold without reference to any special
directions in space?
If we had been thoroughly brought up in the analysis of
vectors we should at once remember that there is a scalar
quantity V2<£ which has no relation to any special direction
at a point ; and at once, if we begin to look for the simplest
relations that may hold between <f) and p,
V2<£ = kp
suggests itself as one possibility.
This then suggests the method of approach adopted by
Einstein towards this gravitation problem.
Here is the universe, mapped out according to the arbitrary
choice of an observer as a space-time field. This observer
perceives or interprets his physical universe as having certain
gravitational properties characterized by the set of quantities
£VS, and as occupied by a certain distribution of moving matter,
with energy and momentum.
His problem is to envisage the possible relations between
the quantities grs and the matter distribution.
He has to guide him : —
(i) The gravitational theory of Newton, extremely satis-
factory as far as it goes, in which the gravitational
field is represented by a single potential satisfying
This at once suggests that the relations sought must be, at
the simplest, differential equations of the second order.
(ii) The criterion that whatever relations are chosen must
be equally valid whatever may be the arbitrary manner
of depicting the universe in terms of space and time.
7*
ioo RELATIVITY AND THE ELECTRON THEORY
With regard to the preservation of the form of the relations
in an arbitrary distortion of the four dimensional map, we note
that the set of quantities g^ will in such an arbitrary distortion
differ from the corresponding quantities say grs, at the cor-
responding point. Likewise their differential coefficients with
respect to the co-ordinates will take new values.
The problem then is to examine what types of relation
connecting the quantities grs and their differential coefficients,
will represent exactly the same quantitative results as the
same relation connecting the quantities g^ and their differential
coefficients with respect to the new co-ordinates.
For the solution, the necessary pure mathematics was
fortunately to hand. We cannot do more here than refer to
its originators.1 Christoffel (1869) attacked this specific prob-
lem, and his work forms the foundation of the necessary
calculus. Riemann, however, with his peculiar genius, gave
significance to the work of Christoffel by showing its intimate
relation to the geometry of surfaces, and by his investigation
of the possibilities of a geometry in more than three dimensions.
This will form the subject of the next section, but we must
acknowledge here the debt due to Ricci and Levi-Civita who
systematized the work of these two pioneers and left it in a
form ready for the physicist to use.
78. GEOMETRICAL — RIEMANN'S CONTRIBUTION.
In 1854 Riemann submitted to the University of Gottingen
a thesis "On the Hypotheses which lie at the Bases of
Geometry," in which he arrived for questions of measurement
in three dimensions at very much the same position as that at
which we have now arrived in problems of motion. In order
1 For an introduction to the mathematical literature of the whole subject, see
J. E. Wright, " Invariants of Quadratic Differential Forms," Cambridge Tracts
in Mathematics, No. 9.
E. B. Christoffel, " Crelle's Journal fur die Math.," LXX., 1869.
B. Riemann, " Uber die Hypothesen welche der Geometrie zugrunde liegen,"
Math. Werke (1876), p. 272.
" Commentatio Mathematica," Math. Werke, p. 370.
The former paper was translated by W. K. Clifford and published in
" Nature," 1873.
Ricci and Levi-Civita, " Methodes de calcul differential absolu," Mathema-
tischeiAnnalen, 54, 1901.
THE GENERAL TKEO&Y OF R'ELAVrVltV 101
to extract the logical essence of geometrical relations, stripped
of all preconceptions which may possibly be based on muscular
sensations of extent, ocular sensations of symmetry and the like,
he seeks to reduce geometry first to an arithmetical basis. This
is exactly what mathematical physics seeks to do. A point in
three-dimension space is for him merely a set of three numbers.
The distance between two points is merely some expression
in terms of the differences of corresponding numbers which
may act as a measure of the separateness of those points.
The only property that he demands of this measure is that it
should vanish when the two points coincide, and should be
essentially a positive quantity if the two points do not coincide
(cf. what has been said above as to coincidences in physical
measurements). The simplest type of expression satisfying
this criterion for two near points he takes to be ds, the positive
square root of a^dxydx^ the coefficients ars being limited
only by the condition that the expression is always positive.
The problem now arises, however, as to what further
limitations upon these coefficients may be necessary as a
consequence of the fact that all the points considered connect
together to form a continuous whole.
As an illustration let us think for a moment of four points
of some two-dimensional surface and their six mutual distances.
If the surface is plane any five of these distances determine the
relative positions of the four points and hence determine the
sixth distance. Thus, if the six distances are arbitrarily as-
signed, the four points will not in general be coplanar.
Extending this, if a surface is defined, and the position of
any point upon it is marked by two variable quantities X, /t,
the distance between two consecutive points being given by
d? = AafX2 + T&dKdp, + Cdp?.
A, B, C being arbitrary functions of X and //,, it will not in
general be possible that the points should piece together into
a plane surface ; though there will in general be an infinite
number of curved surfaces into which they will piece together.
It is possible to write down an expression involving the
quantities A, B, C, and their differential coefficients with
regard to X and /t, the vanishing of which for all values of X
1 02 RELATIVITY AND :fff£- ELECTRON THEOR Y
and /j, is a necessary and sufficient condition that a plane
surface may be so constructed. If this condition is not
satisfied, this quantity necessarily has the same value at corre-
sponding points of all possible surfaces. It is called the
" specific curvature ".
In exactly the same way if we are dealing with a three-
dimensional region, the six distances of four points in ordinary
space are arbitrary and unrelated, but if a fifth point be added
only three of its distances from the other four can be chosen
arbitrarily, that is, a certain identical relation must hold
between the ten mutual distances. As a consequence of this it
is found that an expression ds2 = arsdxrdxs cannot, as a rule,
satisfactorily represent the distance between two points if
they are to piece together into a continuous three-dimensional
construct (a so-called Euclidean space).
Riemann in fact is able to write down six independent
functions of ars and their first and second differential co-
efficients with regard to the co-ordinates which must vanish
if this is to be so. For convenience we will give them his
symbolism : Bj^ (/t, z/, a, p = I, 2, 3). Of the equations
B^o- = o only six are independent.
It is possible, however, from Riemann's arithmetical point
of view, to think of them piecing together into a three-dimen-
sional region which is, so to speak, a curved region in a four-
dimensional world.
In such a region a system of geometry can be built up
using the given expression for the element of distance, just as
we may have a two-dimensional geometry on the surface of a
sphere. Some of the metrical results of Euclidean geometry
will cease to be true. For example, the three angles of a
triangle on a sphere are not together equal to two right angles.
It remains true on a sphere that the angles at the base of an
isosceles triangle are equal, and that two triangles whose sides
are equal, each to each, are congruent, that is, may be brought
into complete coincidence. But neither of these two pro-
positions is true if the triangles are on a surface of which the
curvature is not the same at all points, as, for instance, an
egg-shaped surface.
THE GENERAL THEORY OF RELATIVITY 103
A being whose knowledge was limited to the perception of
objects in such a two-dimensional universe, discovering this
fact of lack of congruence and having no conception of three
dimensions, could only say either that all objects were limited
in their freedom of movement, or else that their dimensions
were altered when they were displaced from one position to
another according to some local properties of the places in
which they were put. Such alteration might reasonably be
interpreted by such a being as due to some field of force. We
who live in three dimensions on the other hand ascribe it
simply to the limitations of the observer's universe.
79. EXTENSION TO A HIGHER NUMBER OF DIMENSIONS.
Our ordinary systems of pure geometry based on a general
notion of absolute direction and absolute length express the
properties of surfaces in a form which is independent of any
particular choice of numerical co-ordinates. In reducing his
geometry to a purely arithmetical basis, Riemann has therefore
to replace the intuitive geometrical notions by a mathematical
proof that the equations developed represent the same truths
whatever be the system of co-ordinates adopted, so that no one
system is to be preferred to another. This is completely
parallel with the Principle of Relativity, which extends this to
all the phenomena of motion.
Having reduced geometry to a numerical basis the way is
clear for an extension of the number of dimensions, and
Riemann proceeds to the discussion of this. Any set of
co-ordinates being x^ xz, x^ x^ and the element of length
being given by
the condition that the elements may fit together into a four-
dimension space is again expressed by the equation BJ^. = o,
but there are now twenty independent expressions instead of
six. If these expressions (B£vo.) do not vanish two interpreta-
tions are open to the observer. The first is that there is a fifth
dimension, and that for some reason his four-dimensional
universe is a curved section of this, the quantities B£v<r being
measures of the curvature. Or on the other hand if he has no
104 RELATIVITY AND THE ELECTRON THEORY
power to think into five dimensions, he will naturally attribute
properties varying from point to point of his four-dimensional
world which will correspond to the existence of physical
phenomena in that world. As Riemann puts it, "we must
seek the ground of its measure relations, in binding forces
which act upon it ".
If they do vanish Riemann shows that there is a system of
co-ordinates in which the element of length is the ordinary
expression
ds* = dx? + dx? + dx* + dx?
or, in geometrical language the measure relations are those of
Euclidean space, x^ x%, xz, x^ being then a set of rectangular
co-ordinates.
80. APPLICATION TO PHYSICAL THEORY, CONDITIONS FOR
ABSENCE OF REAL GRAVITATIONAL FIELD.
Without using any of the geometrical language of the
preceding section, which has been inserted by way of illus-
tration, we may quote this main result of the mathematical
theory.
There are twenty independent combinations of the quantities
grs and their first and second differential co-efficients with respect
to the co-ordinates, the vanishing of which gives the conditions
that the element ds* can by a suitable change of co-ofdinates be
put in the form
ds* = dx* + dy* + de2 - <?dt\
If this is the case, and an observer used these co-ordinates,
a free particle would appear to describe a straight line uni-
formly and light would travel in straight lines — there would
in fact be no evidence of any gravitational effect.
These quantities being denoted by B£va. the equations
B^ = o
express then the condition for the absence of a real gravitational
field.
8 1. EXTENSION TO INCLUDE A REAL GRAVITATIONAL
FIELD.
Now our ordinary experience is that gravitation and matter
are intimately allied with one another, and we shall not expect
THE GENERAL THEORY OF RELATIVITY 105
that we shall be able in the material universe to choose a
frame of co-ordinates which shall entirely transform away the
gravitational field. In the case therefore where the functions
B£v(r do not all vanish, the question arises as to how they are
to be related to the presence of matter in the field.
Looking back to the familiar Laplace equation
.**-
l*z* '
and having come to the conclusion that the gravitational field
must be described in terms of the tensor g^v instead of the
single quantity <£, Einstein looks for a set of equations in
number equal to the number of ^-'s, differential equations of the
second order, which are satisfied in the particular case of
a purely fictitious field, that is when Bj^ = o, but which are
more general in character. These equations must of course
retain their form under any arbitrary change of co-ordinates.
Putting GM, = 2P B^p, (p= i, 2y 3, 4), it is found that the
set of equations
G^v = o
satisfies all the necessary criteria for being a suitable extension
of the equation V2$ = o, the equation satisfied by the gravi-
tational field at a point not actually occupied by matter.
It is shown that if a change of co-ordinates be made whereby g^v
change into g'^vy and GMI/ into G'^.v then the equations connecting G'Mv
with GMV are identical with those connecting g^v with g^v. Such a set
of quantities is said to be a covariant tensor of the second order. Einstein
seizes upon these equations, and takes them as characteristic of a region
not occupied by matter. We shall see later how he was able to make use
of them.
Summary of Mathematical Formulation.
ds* = gndxidxs (i)
b, = grso", s = I, 2, 3, 4
and these equations are solved to give a3 in terms of br we write
as = g*b, . . . (ii)
{pv, X} = g^jafj a] . . . (iv)
106 RELATIVITY AND THE ELECTRON THEORY
BPM<r = \iiff, e} {ev, p} - {iw> e] {e<r, p} + — {^cr, p] - — {pv, p} (v)
G^-B;,, . . (vi)
As always, it is implied that summations are to be affected with regard to
the suffixes which occur twice in all these expressions, viz. :—
r in (ii), a in (iv), e in (v), and p in (vi).
82. THE PATH OF A FREE PARTICLE HAS STATIONARY
LENGTH.
Although Einstein's guess at a possible form of the equa-
tions was actually arrived at along the lines just indicated, it
will perhaps enable the reader to get a firmer hold of the fact
that a set of differential equations expressed in terms of a
special set of co-ordinates in reality embodies a fact entirely
independent of the interpretation in terms of space and time
which is attached to those co-ordinates, if we express them in
a different but equivalent form.
In Newtonian dynamics we know that a particle in a field
in which there is no force acting on it moves in a straight line
uniformly. In the four-dimensional picture of its motion in
which the Newtonian space and time co-ordinates are actually
used, the motion of the particle is represented by a straight
line, that is, a line such that the differences in the space co-
ordinates of two positions of the particle are proportional to
the difference in the time co-ordinate.
We recall the identification of a straight line as the shortest
distance between two points. The length of a line in a four-
dimensional picture is to be defined thus. In the absence of
a gravitational field the interval between two adjacent points
has been defined in terms of the small differences of co-
ordinates, thus —
ds* = - dx* - dx? - dx* + dx*.
Let a series of points be taken along the curve and the
sum of the intervals between each point and the next be cal-
culated. We find the limit to which this sum tends as the sub-
division of the line becomes finer and finer, and we call this
sum the total interval between the terminal points for the
given curve.
THE GENERAL THEORY OF RELATIVITY 107
Denoting this by \ds> it is clear that if the curve is changed,
the length of it is in general changed. But let the curve be
straight, in the sense spoken of above, and then let it be bent
a little out of the straight by giving to each point of it an
arbitrary small displacement. The mathematician can show,
by what is known as the calculus of variations, that such a
slight displacement will, to the first order of small quantities,
leave the total length of the curve unaltered. We say that it
has a stationary value. Further, it is capable of proof that
the straight line is the only curve for which this is true. This
is the equivalent of the shortest length property of the
Euclidean straight line.
Thus the path of a free particle in the absence of any
gravitation is such that \ds is stationary, for any slight varia-
tion from its passage between two given point instants. For
convenience the term ' geodesic' may be used to connote this
property, a term borrowed from the geometry of curves on
surfaces. But we do not conclude that the total interval is a
true minimum. As a matter of fact it is in general a maximum.1
83. THE GEODESIC PROPERTY A RELATIVE ONE.
Consider the picture of the same particle in the same con-
ditions merely by transforming co-ordinates. In the new
picture there is what we have spoken of as a fictitious gravita-
tional field. The particle will freely obey this field, but its
world-line will no longer be straight. But, inasmuch as there
is an exact point-to-point correspondence between the two
pictures, and the separation of two adjacent points ds is
the same in both, the path in the transformed picture will be
also a geodesic.
At this point we recall what has been said of the
equivalence of a fictitious and essential gravitational field.
Thus we pass naturally to the hypothesis that the path of a
particle in any field is such that the total separation of the initial
and final configurations is stationary.
1 Because we have taken dsz = - dx^ - dxzz - dx3z + dx4"
instead of dsz = dx* + dx2z + dx3* - dx4*.
io8 RELATIVITY AND THE ELECTRON THEORY
This brings us to the mathematical problem of rinding the
path for which f/
is stationary as the equivalent of the problem of finding how
a free particle will move in a field of gravitational force
characterized by the tensor gvi.
So far, of course, the precise relation of this tensor to the
ordinary gravitational potential has not been shown. This will
appear later when we take some particular cases.
In the same way, in the complete absence of a gravitational
field, real or fictitious, we come back to the equation of pro-
pagation of a ray of light as
ds = o.
In a purely fictitious field the same equation must be satisfied,
and, in a field which is not entirely fictitious, the assumption that
the same equation holds will satisfy the hypotheses of Einstein.
It remains to be seen whether it agrees with experiment.
84. HAMILTON'S PRINCIPLE OF STATIONARY ACTION.
In passing to the wider field of physical phenomena in
general, we take up the suggestions from various quarters1
that, in the re-thinking of physical theory, the Principle of
Least Action is that which stands least in need of modification.
We have seen in regard to the special Principle of Rela-
tivity that that Principle not only left the action principle
standing — but that the action was an invariant quantity, in fact
the only quantity associated with material systems which had
a value independent of the space-time system of measurement.
We naturally, therefore, look for a quantity which is
invariant under the most general transformation, and, having
found one, examine whether the consequences of treating this
quantity as the action will lead to laws at all consonant with
the known phenomena.
The researches of Riemann give us at once a quantity that
is invariant, namely, the quantity G = 2 ^ g^v GMV, the sum-
mation being made over all possible sets of values of /i, v.
This is a quantity defined for each point of the four-dimen-
sional world (that is, of time and space).
1 Particularly by Sir Joseph Larmor, " ^Ether and Matter," and elsewhere.
THE GENERAL THEORY OF RELATIVITY 109
85. NOTE ON THE GEOMETRICAL SIGNIFICANCE OF G.
Here the work of the geometers, referred to above (§ 78),
is of fundamental importance. Gauss showed that if a surface
of inextensible but flexible material be deformed — the geo-
metrical conditions being merely that the distance between
two neighbouring points does not alter, then throughout all the
changes of shape that the surface may undergo there is one
quantity associated with each point of the surface that does not
change. He called it the specific curvature ; in the language of
the geometry of surfaces it is the product of the curvatures of
the principal sections of the surface at the point. Its value is
determined simply by the nature of the surface, and is, there-
fore, quite unconcerned with any particular mathematical
mode of defining the position of a point on the surface.
This can be generalized, following up the lead given by
Riemann to any number of dimensions. For our purpose
here it is sufficient to know that in any four-dimensional con-
nected region built up of fixed measured elements of fixed length
there is an invariant quantity associated with each point — we
may call it the generalized curvature — though that phrase has no
meaning except in relation to a hypothetical five-dimensional
space in which the four-dimensional region is drawn as an
ordinary surface is drawn in three-dimensional space. This
quantity depends only on the mutual relations between the
lengths of neighbouring elements, and not upon any particular
analytical method of specification. It is thus that the invariant
quantity G is reached.
86. INVARIANT ELEMENT OF INTEGRATION.
In §62 in which the electro-magnetic field was being
considered in relation to the principle of least action, it was
remarked that under the Lorentz-Einstein transformations, the
element of integration in the four-dimensional world was
invariant. This was clear because those transformations
consisted merely in regarding the same element from a set of
co-ordinates obtained from the old by a mere turning round
of the axes.
In the present general theory, in order to compare the
i io RELATIVITY AND THE ELECTRON THEORY
contents of the given element as portrayed in two separate
sets of co-ordinates, we have to remember that we not only
turn the axes about but we may stretch and warp the picture.
We find, however, that the element of the four-dimensional
space is increased in exactly the same ratio that Jg is
diminished ; 1 so that there is an invariant element Jg ^ID.
87. HYPOTHETICAL EXPRESSION FOR ACTION.
We thus arrive at an expression which for any given region
of the four dimensional world is an invariant, viz : —
The subject of integration here is simply a function of the
quantities grs and their first and second differential coefficients
in regard to the co-ordinates.
For a field in which gravitation can be completely trans-
formed away, a field which has previously been called a
Galilean field, the gravitation being as we have said com-
pletely fictitious, the value of G is zero.
The conditions that this integral shall not change its value
(to the first order) when arbitrary small changes are given to
the £-'s, are found to be exactly that set of differential equations
upon which Einstein first alighted as the simplest with which
the repertory of the pure mathematicians could furnish him,
viz : —
GM, - o.
Thus we are able to put Einstein's theory in this comprehen-
sive form, that there is a numerical quantity which can be
calculated out of the distribution of the gravitational field, the
value of which is entirely independent of the choice of the observer
as to the means by which he describes the positions and motions
of the material bodies ; and that the distribution of the field is
always such as to make this value stationary as compared with the
value of any slightly differing field.
Lg stands for the determinant
ll> ^12' ^13' g
THE GENERAL THEOR Y OF RELA TIV1TY 1 1 1
88. THE MATTER TENSOR.
Returning to the explicit differential equations adopted by
Einstein, it is necessary to contemplate what happens at points
at which GMV does not vanish. If G^v = o is the equation
characteristic of the absence of matter, the expression GMV must
measure the relevant properties of matter. Our conception of
those properties may have to be revised to meet the new
theory. We have already learned the interdependence of
energy and momentum, and the relative nature of force and
stress. We can therefore only say at present that there must
be a relation of the form
where 0^ is some tensor which is characteristic of the state of
matter at a point. The fact that it is a tensor will ensure that
the equation represents a physical relation independent of the
choice of the system of co-ordinates.
We speak of 0MI, as the matter tensor or the energy tensor.
Its relation to the motion of the matter and the ordinary con-
ceptions of the properties of matter may for the moment be
left ; but it should be emphasized that the above equation
does no more than identify certain aspects of the gravitational
field with the matter. It associates together the curvature of
the measure system as defined by the quantities GM|/ and the
life history of material bodies.
89. THE LAWS OF CONSERVATION IN EINSTEIN'S THEORY.
In the old mechanics, three laws stand out prominently,
the laws of conservation of mass, of momentum, and of energy.
In Einstein's early theory of relativity, and indeed prior to that,
the conservation of mass ceased to have validity ; we have seen
that the idea of constant mass is intimately associated with that
of constant internal energy (p. 80).
But the laws of conservation of momentum and energy
remain in Einstein's theory ; in fact they can be shown to be
an immediate consequence of his law of gravitation. The
relation of these laws of conservation to the general laws of
ii2 RELATIVITY AND THE ELECTRON THEORY
the field may be put thus. Starting from the expression for
the action,
an integral through all space of a function of g^v and its
differential coefficients, two conditions have to be satisfied.
(i) A is invariant when a change of co-ordinates is made
with a corresponding change in g^v.
(ii) SA = o when arbitrary changes Sg^ are given to the
tensor^.
Now it is quite easy to see that the conditions that (ii) may
be so, include the conditions for (i). The condition for (ii) is
in fact Einstein's equation
GMV = o.
If G were not known to be invariant these would be ten
independent equations, but if G is invariant there are four
expressions derivable from the ten quantities G^v which vanish
identically whether G^v = o or not
Thus when G^v is interpreted as characterizing the state
of matter at any point, there is found to be a quartet of
relations satisfied by these quantities. The form of these
relations corresponds precisely to the form of the equations
usually taken as representing the laws of conservation of
energy and of momentum.
Thus on Einstein's theory these laws are an immediate
consequence of the invariance of the action, quite apart from
the principle of least action. l
1 For a different expression of the intimate relation between conservation
and relativity, see Eddington, " Space, Time, and Gravitation," p. 139.
CHAPTER VIII.
THE VERIFICATION OF EINSTEIN'S EQUATIONS.
90. CONSEQUENCES OF THE NEW THEORY.
WE come now to an examination of the two results of
Einstein's theory which have been so strikingly verified.
Newton's simple law of the inverse square for gravitation
together with his scheme of dynamics had proved to account
with great accuracy for all the planetary and lunar motions
with all the irregularities observed in those motions,1 except
for one or two outstanding instances, of which the most marked
was that of the motion of the perihelion of Mercury. After
allowance for all known causes there remained an unexplained
progression at the rate of 40*2" per century.
In 1915 Einstein was able to show that his new theory
would in the first place give all the results of the Newtonian
theory ; but further he found that a correction had to be added
which in the case in question amounts to a progression at the
rate of 43" per century. In all other cases the correction is
negligible and leaves the agreement between theory and
observation unspoiled.
It seems sufficiently surprising that generalities such as
those which have been described in the preceding pages should
suggest any precise numerical result at all ; but that they should
succeed in explaining the one outstanding discrepancy within
the degree of accuracy of which the observations are capable
is sufficient to bring the theory at once into the front rank.
The second verification of Einstein's theory was in respect
of its prediction that light rays from a star would be deflected
if they pass near to the sun, thus producing a displacement
1 The classical instance is the prediction of the existence of the planet Neptune
from a consideration of the perturbations of Uranus by Prof. J. Couch Adams
and Leverrier.
113 8
114 RELATIVITY AND THE ELECTRON THEORY
which should be measurable. This we shall consider in § 93.
But first we must consider in detail Einstein's discussion of
the field of gravitation round the sun.
91. THE FIELD ROUND THE SUN TREATED AS A SINGLE
GRAVITATING PARTICLE.
The law which Einstein proposed for the gravitational field
in free space is, as has been said,
G^,, = o.
This is a set of ten equations connecting the quantities g^v and
their first and second differential coefficients.
We begin to ask what results from this differential equation
if the field is associated with a single centre, thinking of the
sun as a mere particle or as a spherical distribution of matter.
Two characteristics may at once be laid down : —
(i) The field is constant, that is does not vary with the time.
(ii) The sun's gravitational field as measured by an observer
on the sun, will be symmetrical round its centre.
The first condition requires that the quantities g^v are all
independent of the time.
The second means that the expression
must be built up from quantities which have no special relation
to any particular direction through the centre of the sun.
If we use spherical polar co-ordinates the only such
quantity in addition to rt the distance from the centre and
quantities derived from it, is the element of distance between
two neighbouring points on a sphere whose centre is at the
centre of the sun. The square of this distance is known to be
r\de* + sin20 aty2).
Thus the two conditions limit the form of ds to that given by
ds* = f(r) dr* + $(r) (d&* + sin20 d<f) + itfr) dp.
But again, in view of the general Principle of Relativity,
there is nothing to say what is the proper way to measure the
distance from the centre of the sun ; so that we will say quite
arbitrarily that r shall be so measured that <£(r) is - r2.
Thus ds* is now
f(r) dr* - r>(sin20 dtf + <#2) + ^(f) dt\
THE VERIFICATION OF EINSTEIN'S EQUATIONS 115
Or, g-u = f(r\ g^ = r* sin20, gzz = r\ g,, = f (r),
and all the other g^v are zero.
It remains to write down the explicit form of the equations
G^ = o
with these simplifications ; when this is done it is found that
it is possible to find without ambiguity the form of the functions
f(r) and ty(r) ; in fact we find that
W) = i - A/r and f(r)= - (i - A/r)-1
where A is a constant of integration. A is in fact the con-
stant which shows how the field differs from a field of no
gravitation ; for if A were zero we should have
d£ = - dr* - r2 (sin2 0d$* + dO*) + dt\
which in ordinary Cartesian co-ordinates would become
- dx* - dy* - dz>> + dt\
Thus, except for the evaluation of the constant A, there is
a unique solution of the gravitational equations subject to the
conditions of permanence and symmetry. This solution applies
only to the region from which matter is absent. It would
hold in the region outside a spherical distribution of matter,
but not within the matter itself.
The constant A clearly has something to do with the sun's
power to produce a gravitational field ; and we shall see that
it corresponds to 2ym, where m is the mass of the sun, and 7
the constant of gravitation.
92. THE MOTION OF A PLANET IN THIS FIELD.
The next step is the determination of the motion of a
particle moving freely in this field ; a motion which is deter-
mined by the condition that ds has a stationary value. The
world-line of the particle is a shortest
distance in the four - dimensional
map.
We take any arbitrary world-line
from a point O to a point Q.
fi> FIG. 10.
Let s = \ ds, P being any
Jo
point on this line. Let an adjacent world-line be drawn, the
point P' corresponding to P being such that its co-ordinates
n6 RELATIVITY AND THE ELECTRON THEORY
differ from those of P by 8r, 80, 8$, 8tt these being arbitrary
functions of s, subject to the condition that they vanish at the
ends.
Then the increment in [*ds when evaluated along this
J o
varied line instead of the original line is
= SJFtts
where F2 = ifrff* - r*\ty - ^(sin2^'2 + 0'2), accents denoting
differentiations in regard to s. According to the ordinary
method of the calculus of variations1 this change will vanish
to the first order in 8t, 8ry 80, 8<f>, whatever functions of s
these may be if
c)F <>F
From (iii) and (iv) ^-77 and ^ are constant along the re-
quired path. Also by the definition of ds, F = I all along
this path.
Thus we obtain
^sin2 0<f>' = h
and tyt = k
h and k being constants.
JThus
Also 5/ = 5=
ds ds
so that, integrating by parts,
if Sr = o at both ends of the path. Similarly for the other co-ordinates. Thus
3F ''/'
ar -i ^Is
-
ds W
THE VERIFICATION OF EINSTEIN'S EQUATIONS 117
Again Fg -
or, since F = i and therefore — - = o
as
d/DF\ d^
Also F — = - r>sin0 costfcp
. '. r'sintf cos#$'2 = r*B" + 2rr'ff.
Now, at any specified moment the particle is moving in a
certain plane through the origin. Let the axis from which 9
is measured be taken normal to this plane. Then at this
moment 0 = - or cos# = o, and 6' = o.
2
Hence 0" is zero, and 6 remains constantly at the value
7T/2. Thus the orbit remains continually in this plane.
Instead of integrating equation (i) we use F = I, which
must be consistent with it.
Putting 6 = 7T/2 and tyf = k this becomes
Substituting r^tf = h
this gives rz = tf - I + A/r - #Y^ + AA2/^-
In the absence of the last term this equation would have
been exactly the Newtonian equation, except that here r is
drjds instead of dr\dt. As far as the shape of the orbit goes,
however, this makes no difference, since it is obtained by
eliminating ds or dt between the last two equations to give
an equation connecting r and <£> only.
Writing A = 27^, h? = 7^/, and k2 - I = ym/a the last
equation becomes
The corresponding Newtonian equation is
where S is the mass of the sun.
Thus we are able to identify m with the mass of the sun.
ii8 RELATIVITY AND THE ELECTRON THEORY
93. NUMERICAL CALCULATION OF THE CORRECTION TO
NEWTONIAN THEORY.
The value of ym (taking the unit of length to be i kilo-
metre, and the unit of time — — th second, so that the velocity
300000
of light is unity) is given by the equation
periodic time T = 2ira */ /J^m
which from the earth's orbit gives
ym = a3/47T2T2 = 1-47.
Thus, in the orbit of Mercury, since r is of order io8 (kilo-
metres) ym/r is of the order io~8.
Thus ty differs from I by the order io~8, and the extra
non-Newtonian term (in our equation) is of the same order
compared with the other terms on the right-hand side. We
may say, therefore, that to that order of approximation the
Newtonian calculations are correct.
We proceed to find the effect on the orbit of the correcting
term, taking account only of the first order in ^r.
The equation for the orbit eliminating s or t is
/ I 2 / 2ym/\
ym( — _ + - — -- H '-^ }
\ a r r* r6 J
or putting as usual u = i/r.
'i) - I/a/ + 2ul - uz -
This is to be integrated to give the polar equation of the orbit.
Here we consider only the first order effect on the apse
lines due to the additional term.
Write the equation
u (a -«)(«- ft (B - 2ymu)
where a and /3 are the corrected values of u for which
dujd^ = o, that is, the new apsidal distances. B is given by
B + 2ym (a + /S) = I,
and, therefore, is only slightly different from I.
During the motion u is confined between a and {$.
Thus = _ .„
du N/\B(a - u) (u - /3) \ B
a - u)(u- ft \ B
THE VERIFICATION OF EINSTEINS EQUATIONS 119
We integrate between u = a and u — /3 to obtain the
angle c/> between the apse lines. In the Newtonian case this
anle is TT.
pj , _
e~
I f du _ ym r _ u du
- f -
TT
=
Putting in the value of B in the first term, and neglecting
= TT {i + ym(a + fi)} + — ym(a
2
= TT{I +
In the second term, which is small, we may put for a and ft their
approximate values a-j(i - e) and a/(i + e) of the Newtonian
orbit so that
(a + ft = 2/*(l - *2) = 2//
Thus <£ = TT(I + 37^//).
The angular distance from apse to apse exceeds TT therefore by
the small fraction 3ym/t of itself. Thus the apse line moves
forward in the plane of the orbit.
For Mercury we have
/ = 5-55 x io7 kilometres, and ym = 1-47 kilometres,
giving 37^// = 7-95 x io~8.
The time of one revolution of Mercury is 87-97 days.
With these figures the rate of advance of the apse line
works out as
42-9" per century.
The outstanding discrepancy to be explained was
40 -2" per century.
94. THE DEFLECTION OF LIGHT IN A GRAVITATIONAL
FIELD.
The law which has been suggested for the propagation of
light is ds = o. Taking the expression obtained above this
gives
. 120 RELATIVITY AND THE ELECTRON THEORY
Confining our attention to a region in which ym/r is small we
can rewrite this equation
(*)' t ,,,{«io,,(5)- + g)'} , (, . as)'
where ^ = r - //, ; so that if we plot the path of the ray of
light by means of the co-ordinates rlt 6, <f>, we find that its
2ym
velocity is I - -
>-i
In this picture then the light will move as in a medium
whose refractive index u is (i — — ) or I 4 — — . This in-
\ r, J r,
creases as rt diminishes so that the rays will be concave to the
origin.
To calculate the complete deflection of a ray coming from
a long distance and travelling to a great distance, we proceed
thus : if / is the perpendicular from the centre on the tangent
the ray travels so that pp = const.
Thus p(i + 2^) = const.
if R is the distance of nearest approach to the origin.
This gives R(* + *X**/R) = j + ?7!?
or squaring
This can be identified with the polar equation of a hyper-
bola with respect to the focus, viz. : —
£2 2a
? =" l + ^
by putting a = 2ym
and £2 = R2(i -f
or b = R(i +
The total deflexion of the ray is the angle between the
asymptotes, viz : — _ 1 a
"2 tan -y
o
which to the first order is
2tan~'ir - TT
THE VERIFICATION OF EINSTEIN'S EQUATIONS 121
On drawing the corresponding curve with r as the radius
vector instead of rL it is easily seen that it is a curve with
the same asymptotes, so that the deflexion in the system of
co-ordinates commonly used is that calculated.
Taking ym = I '47 kilometres and R = 697,000 kilometres
we find for the predicted deflexion of a ray passing* close to
the sun's surface a deflexion of 1 74".
This figure was strikingly verified by the results of the
photographs of stars taken in the region round the sun at the
total eclipse in May 29, 1919. An account of the details is to
be read in Prof. Eddington's " Space, Time, and Gravitation,"
Chapter VII. Some further discussion of the interpretation
of the result will be found in the next chapter (p. 121).
95. THE SUGGESTED SHIFT OF THE SPECTRUM BY
GRAVITATION.
The prominence given to the interval between two events
as an invariant quantity with a physical significance1 has led
to a much-discussed suggestion that the radiation from a given
atom will be altered in period if the atom is moved into a
gravitational field of different intensity. The suggestion arises
thus.2 Let ds be the interval measure between the beginning
and end of a single vibration of a radiating atom, as measured
in a given set of co-ordinates, at a certain place in a gravitational
field. By variation of the system of co-ordinates, we shall see
the same atom in an apparently different field of gravitation,
but ds will remain the same. Thus it is suggested that ds will
be a permanent quantity, characteristic of the vibration of this
atom, and that all identical atoms will give the same vibration
interval ds whatever the field in which they were placed.
If this were so, if two identical atoms were at rest in two
different places, their time-space co-ordinates being respectively
(x, y, z, /), (x\ /, /, /), and the gravitational tensor being
gp, and^-'MJ, for the two atoms respectively, we should have
ft**.'-***!1
on putting &r = 8x' = o, etc., to express that the atoms are at
rest in their respective fields. Thus the corresponding intervals
1<SM**
2 See Eddington, " Space, Time, and Gravitation," p. 128.
122 RELATIVITY AND THE ELECTRON THEORY
of time, for the two vibrations of the respective atoms, are in
the ratio Jg^ : *Jg\±. Taking the values of g^ at the sun's
surface, and at the distance of the earth from the sun in the
field, as given in § 90, this ratio differs from unity by about
•000002, which would correspond to a shift of light of wave-
length 4000 A by an amount -008 A towards the red end.
Such evidence as there is, is apparently against the ex-
istence of such an effect.
But it is important to note that the argument is based on
a hypothesis which has not been used in connection with the
two phenomena which have confirmed Einstein's theory ; and
that is that nature itself will provide a measure of the interval
between two events. It is true that in the transformation
from one co-ordinate system to another the interval between
two given events remains fixed. But that is not the same as
saying that, in any one co-ordinate system, we can say of
two pairs of events, defined by the vibrations of two widely
separated atoms, that the intervals are the same.
Thus the failure to find evidence of this suggested phe-
nomena does not weaken the general confirmation of Einstein's
theory. Not only so, but in the next chapter (§ 105) we
shall see that a generalization of Einstein's theory is made
in which the interval between two events is an indeterminate
multiple of the quantity (g^d^x^, so that it becomes im-
possible to say that this is a universal quantity for the interval
between two successive vibrations of an atom.
96. CONCLUSION.
Beginning then with an attempt to satisfy the meta-
physician, Einstein has succeeded in making an analysis of
physical phenomena which is in most marked agreement with
observation. It is therefore an analysis which has to be
accepted as the most comprehensive hitherto achieved.
Let us now summarize the position, especially, in regard
to the concepts of space and time, emphasizing the point at
which the new theory goes further than the special principle
of 1905.
Like that principle it maintains the interrelation of the
THE VERIFICATION OF EINSTEIWS EQUATIONS 123
two aspects of the universe, spatial and temporal ; it also
retains the possibility of assigning a definite numerical value
to the "interval" between two events, a value which is the
same in all systems of measurement
One point which the new theory emphasizes is that it need
not be possible for these numerical measures of the intervals
between all events to be fitted together to satisfy the require-
ments of Euclidean geometry.1 Thus, when a background or
set of co-ordinates has been chosen, and a time scheme has
been adopted in an arbitrary manner, the conceptual space
intervals may not fit together like ordinary Euclidean ele-
ments of length. But, we must remember that the relation
of these conceptual space intervals to measured lengths has
yet to be discussed (see pp. 126 fT.).
Here it is wise to warn the reader against possible con-
fusions in language. Some would say that the last sentence
may be paraphrased thus "Space is non-Euclidean " — implying
that for them Space is nothing more than an aspect of the
metric relations between material phenomena — it is not a
background of co-ordinates, it is not the aggregate of all sets
of values of three parameters.
This is a logical attitude. If it is taken, the quantities
g^v are characteristic of and indeed define the relation of the
set of co-ordinates chosen to the complex of physical changes.2
But it is most important to keep it always in mind that
the word ' space ' is now only a convenient term to cover the
metric relations of the physical world.
On the other hand, there are those who would use the word
1 This is not even true in the case of the special Principle of Relativity ; for
there the line element is given by
d& = - dxz - dy* - dz* + dt*
and if x, y, z, t are all real quantities the related geometry is not Euclidean.
Only when t = T >J( - i), r being real, do we arrive at a quasi-Euclidean geometry.
In the actual universe it is generally possible to join two points by a line of zero
length, a distinctly non-Euclidean property.
2 This is presumably what Prof. Eddington means when he says (" Report,"
p. 22) that " the values of the ^'s express the metrical properties of the space
used". Here clearly "space" means co-cordinate system. Later, however,
we read, " When he speaks of space, he means the space revealed by measure-
ment, whatever its geometry" — (" Space, Time, and Gravitation").
124 RELATIVITY AND THE ELECTRON THEORY
t space ' in such a sense that by itself it has no metric properties.
It cannot be said to be either Euclidean or non-Euclidean.
Given the complex of phenomena we are at liberty to choose out
any arbitrarily moving point and say that its course is a straight
line defined uniformly — or we may take any curve whatever in
the four-dimensional picture and call it straight We may take
any fourfold meshwork as defining a set of parameters. These
may be taken as ordinary Cartesian co-ordinates — the picture
may be strained, so to speak, until the parametric surfaces are
flat. There is a Euclidean geometry in this fourfold back-
ground, but that geometry does not correspond with the
geometry of measured motion. The Euclidean element of
length is not the measured element of length. Over and
above the frame of reference we require to know the values of
the coefficients g at each point. From this point of view, to
the individual observer who has chosen his frame of reference
— once for all — the coefficients g appear as properties of the
physical world relative to his frame of reference. Here we
are nearer to the Newtonian position ; and provided we re-
cognize the equal validity, that is, the complete relativity, of
all space-time partitions of the physical world, there is nothing
to hinder.
But the former position is in one respect to be preferred.
For it recognizes the physical reality as one, and the mental
pictures of it as many, subjective and relative. The latter
position requires one to think of quantities such as g^v as
representing metrical properties of the physical world, although
their values only partially depend upon physical phenomena,
and indeed are largely arbitrary.
The theory recognizes no physical significance in any
equations which are dependent for their validity on the choice
of a system of measurement. In that sense it is a theory of
the " absolute " physical world, the world absolved from indi-
vidual views of it. The relativity is characteristic of the view.
CHAPTER IX.
FURTHER CRITICISM AND GENERALIZATION.
WEYL'S THEORY OF ELECTRICITY,
(The reader will have realized that we are dealing now with an essentially
mathematical theory, and it cannot be hoped that this chapter will give any
other impression. The main conclusions have been italicized. I have to
thank Prof. H. F. Baker for valuable criticism of this chapter.)
97.. RECAPITULATION.
IN the preceding chapter the order of the argument has been
mainly historical ; the purpose has been to give a straight-
forward account of Einstein's method and results.
Before concluding, it is worth while however to re-
capitulate and to see the general lines upon which a radical
reconstruction of physical theory must proceed. We shall see
that even Einstein's generalized relativity is not the last
word, and an attempt will be made to indicate the lines along
which Weyl has shown that it is possible by a further step to
give to electricity as natural a place as that accorded to
gravitation in Einstein's theory. Einstein himself has shown
that it is possible to generalize the fundamental equations of
electrical theory so that they have an invariant form, and are
therefore consistent with the Principle of Relativity in the
sense in which he uses the term. But in his treatment the
connection between the gravitational field and the geometry
of the measure system is so intimate, while that between the
electrical field and the measure system is so remote, that the
fundamental place accorded to electricity in modern thought
seems to be denied. In Weyl's theory it is completely
restored, and gravitation and electricity rank together as two
fundamental properties of matter, forming together the basis
of all natural systems of measurement.
98. RECONSTRUCTION.
The universe of phenomena is now conceived to be laid out
as a four-dimensional whole. Corresponding to our ordinary
125
126 RELATIVITY AND THE ELECTRON THEORY
conception of a point of space at a definite instant of time,
we have in this four-dimensional whole a marked point or
position, which to save confusion it is better to call an event.
The history of a particle is recorded as a continuous one-
dimensional sequence of such events, or a curve.
For the purpose of distinguishing between events we
introduce a set of four variable quantities (x^ x^ x^ #4) ; to
each event corresponds a definite set of values of these. Two
events are neighbouring events if the values of these co-
ordinates differ only by small quantities.
Our object now is to search out definite relations between
these events, to describe by comprehensive formulae the
nature of the sequences which we are able to observe. This
is the work of the mathematician, who must accordingly
develop his machinery.
99. THE NATURE OF OBSERVATIONS AND OF
MEASUREMENT.
It has several times been remarked above that the only
precise observations are those of coincidences.1 The physicist
constantly seeks to eliminate from his experiments the ap-
plication of personal human judgment to the estimation of
intervals of time or space. He makes the photographic plate,
the micrometer screw, the galvanometer scale do the work
for him. Thus the work of observation is reduced to the
recording of the fact that a certain event in one chain of events
coincides with a certain other event in another chain ; or, in
other words, of the fact that a certain event is common to two
particular world lines. The work of theory consists in dis-
covering the relations that hold between such observations.
But, it may be asked, granted that all operations of
measurement may be reduced to observations of coincidences
between marks on scales and on the bodies under measure-
ment, between spider lines and optical images, and so on,
what is the relation of the intervals of length and time as
measured to the co-ordinate numbers which have been
introduced merely for the purpose of distinguishing one
1 See §§ 24, 43, 72.
FURTHER GENERALIZATION (WEYL'S THEORY] 127
event from another? What meaning in fact is left for such
a term as " a measured interval of length ".
If we analyze elementary methods of measuring distance,
we come back to the calibration of a scale. For the purpose
of doing this, we have to take a certain body, which is con-
ceived to be permanent in configuration (though what this really
means it is extremely difficult to say); we bring a certain
segment of this body between two marks into coincidence
with successive segments of the scale, and mark off the lengths
with which it coincides each time. These lengths are called
equal because they coincide in turn with the same segment
of the trial body. The scale itself is conceived also to be per-
manent in configuration. Thus a measuring rod is prepared.
Thus practical measures of length rest on the conception
of an ideal " rigid" scale. But all that can be said in defini-
tion of the term " rigidity " is, that experiment shows that
we can obtain bodies which give approximately consistent
measurements. Two segments which have once coincided
may be expected to coincide again if brought together. There
is no absolute standard by which it may be said of any one
body that it always has and will have the same configuration.
We can hardly say more than this, that rigidity is the common
property of those bodies whose mutual measurements are
always and everywhere consistent, but that such bodies do
not, in fact, exist.
100. FURTHER DISCUSSION OF THE INTERPRETATION OF
THE ECLIPSE OBSERVATIONS.
Now, to make discussion concrete, let us consider in detail
the interpretation of the actual observations made by the
Eclipse Expedition. These have been said to confirm
Einstein's theory. In what sense is this the case ? What are
the actual steps taken, first in the application of the theory in
this concrete instance ; second, in the practical measurements ?
In § 91, it was said that of all conceivable co-ordinate
systems, only one is consistent with the two hypotheses : —
(i) That the gravitational field does not vary with the
time;
128 RELATIVITY AND THE ELECTRON THEORY
(ii) That the field is symmetrical round the sun as centre
(By symmetry it was implied, that if the space co-
ordinates are treated as Euclidean the gravitational
tensor depends only on the distance from the centre.)
On the basis of this set of co-ordinates a picture of the
universe was set up. The paths of the rays of light were
marked out in this picture on the hypothesis that the equation
ds = o represented the law of propagation. It was then found
that these rays were curved (in the picture) to a greater or less
degree according to the distance at which they passed the sun.
The crucial test is : Can this curvature be detected ? Let
it be emphasized that the curvature is relative to a background
of conceptual co-ordinates. The theory proceeds to map out
the photographic plate, associating with each point of it values
of the Einstein co-ordinates. The displacements of the images
predicted by the theory are stated in terms of changes in the
theoretical co-ordinates of the images. In the actual observa-
tions, photographs were taken of a certain group of stars,
and the relative positions of the images on the plate were
compared with the positions of the images of the same stars
photographed at a time when the rays did not pass near
the sun. Measurements experimentally taken by means of
a micrometer measuring-machine showed certain changes
in the relative positions as plotted in terms of readings on the
micrometer. Theory t on the other hand, predicts that there
will be changes in the relative positions as plotted in terms of
the conceptual scale of co-ordinates.
Until we have assured ourselves of the relation between
the micrometer measurements and the numbers, which for con-
venience we may call Einstein's co-ordinates, we cannot know
what relation to expect between the changes in the positions
of the images in the co-ordinate system and the measured
displacements.
As a matter of fact, the measurements of the plates showed
a surprising agreement with the predicted displacements in the
conceptual picture. What then may we conclude ?
We are tempted to say at once — the theory is correct.
But there is more than this. It is just conceivable that the
FURTHER GENERALIZATION (WEYLS THEORY} 129
theory might be wrong, but that a further discrepancy between
conceptual co-ordinates and measured positions might bring
the prediction again into agreement with observation. This
seems unlikely in a high degree — so that our conclusion is
that there is a considerable probability
(i) That Einstein's conceptual picture of the phenomena
is a consistent one, and
(ii) That his conceptual measures of distance by co-ordinate
numbers are in accord with the experimental
measures. To this point we will return shortly.
101. RESUME OF NEWTONIAN CONCEPTIONS.
Let us before going further see how Newton's stock of
conceptions stands in relation to what has just been said.
Newton postulates an absolute time and space ; but we
must do him justice and admit that, his own definitions not-
withstanding, he does not anywhere assume the existence of
an absolute standard apart from the phenomena themselves.
His space and time are a conceptual scheme employed for the
purpose of describing the relations between phenomena.
On this conceptual background of time and space co-
ordinates he elaborates his picture of the motions of bodies ;
he is thus able to predict which point of his conceptual
space corresponds to a planet at any moment of his conceptual
time; he can express the space-co-ordinate numbers of the
planet in terms of the time co-ordinate number. When this
is done it is found that, except in the long standing discrepancy
in the motion of the perihelion of Mercury, there is precise
agreement, within the limits of accuracy to which we can
rely upon our measurements, between the positions as pre-
dicted by the conceptual co-ordinate numbers and the co-
ordinate numbers actually obtained by measurement. That is,
astronomical observations confirm the suggestion that the
partitioning of space by means of measurements with a so-
called rigid body agrees with the partitioning by means of
Newton s conceptual co-ordinates. This means further that
it is confirmed, indirectly it is true, but none the less distinctly,
9
130 RELATIVITY AND THE ELECTRON THEORY
that the " rigid bodies " involved, scales and micrometer
screws, occupy exactly the same volume and shape as described
by the conceptual co-ordinates, no matter where they may
be placed, whether they are moving or at rest.
Thus the success of Newton's theory, as far as it goes,
confirms the belief that there is a conceptual space-time
system of measurement in which —
(i) Newton's laws of motion hold ;
(ii) Newton's law of gravitation holds ; and
(iii) The conception of a rigid body is tenable.
In other words, these three hypotheses are mutually consistent
in the light of experiment. In itself this conclusion may seem
very meagre, and merely a repetition of familiar truths in a
different form ; but by stating them in this way we avoid
Newton's unphilosophic postulation at the outset of a self-
existent absolute space and time ; we grant the a priori
relativity of all space and time measures ; but at the same
time we recognize that the relations existent within the realm
of phenomena determine what conceptual space-time system
we may most conveniently adopt.
102. THE RIGID BODY: THE PHOTOGRAPHIC PLATE.
In the analysis that has now been made the co-ordinate
system appears merely as a mathematical machinery for com-
paring certain perceived relations between phenomena.
Among these phenomena are the motions of the planets
and the propagation of light. But we must now include as a
phenomenon of perception the existence of the approximately
rigid body. That is, there is no longer supposed to be any
such thing as an ideal or conceptual rigid body. It is simply
a matter of experience that in a certain co-ordinate system,
chosen for the simplicity which the equations of motion take
relative to it, there is a large class of actual bodies which
occupy approximately the same space in the picture wher-
ever they may be placed, and at all time. The " ideal rigid
body" would be nothing more than a body which did this
exactly ; but to speak of such a concept is simply to give a
name borrowed from concrete experience to a purely numerical
FURTHER GENERALIZATION (WEYUS THEORY] 131
conception. There is no such thing, therefore, as " measure-
ment by the ideal rigid body ". The phrase means nothing more
than the comparison of certain perceived phenomena through
the mediation of a conceptual co-ordinate system. But the
rigid body of experience is essentially a complicated dynamical
system. Its property of permanence is of the same kind as the
permanence of the solar system as a whole.
Thus we have to recognize that the measuring of the eclipse
photographs by the micrometer might quite possibly give re-
sults differing from the measures by means of the co-ordinate
numbers at which theory arrives.
But just as the system of co-ordinates which is evolved by
Einstein as most appropriate to the gravitational field round the
sun differs only slightly from that obtained by Newton in its
relations to the planetary motions, so it differs only slightly in
its relation to the dynamical system which constitutes the
photographic plate. It is in fact a matter for observation or
experimental verification that a rigid body, in the ordinary
sense, a plate of glass, for instance, or the micrometer screw
does occupy approximately a constant region as marked out
by Einstein's co-ordinates, so that the calibration of the plate
by the screw agrees very approximately with that by means
of the co-ordinates.
Thus the agreement of the theoretical displacement ob-
tained by Einstein with the actual measurements is as much a
confirmation of this property of the rigid body as it is a veri-
fication of Einstein's theory of the nature of gravitation. It is
a confirmation of the mutual consistency of all the hypotheses,
but not strictly a demonstration of any one of them.
103. THE MATHEMATICAL MACHINERY FOR DESCRIBING
THE ORDER OF A CHAIN OF EVENTS.
The method of the mathematician is first to devise a means
of describing in terms of numbers the order or arrangement
of a one-dimensional series of events. If it were a discrete
series, the work would be merely one of enumeration. But
all possible events form for us a continuum, and this requires
that to all the events ranged as they are in order along a
9*
132 RELATIVITY AND THE ELECTRON THEORY
world line we shall make correspond the values of a con-
tinuously increasing number. This is done by assigning a
measure to the separation of two adjacent events. In this
connection an infinitesimal calculus is introduced.
Riemann in his discussion of the bases of geometry at
once introduces as the 'measure of separation of two neigh-
bouring events the quadratic form ds1 = g^dx^dx^ dx^ being
the excess of the co-ordinates of an event P' over those of a
neighbouring event P. This is the simplest expression that
can be devised which will vanish if P' coincides with P, and
which may be positive for all values of the ratios of dx^ : dxv,
that is for all events P' in the neighbourhood of P. Here g^v
are quite arbitrary. The linear form f^dx^ is simpler, but is
excluded inasmuch as the changing of sign dx^ would change
the sign of the whole expression, and therefore lead to a
quantity which would express the separation of two events in
some cases as negative.
104. GENERALIZATION.
But Riemann's method is not the most general that can
be adopted for the purpose we have in view, namely, the
description of the order of events along any arbitrary world
line. It is true that taking any world line through a point P,
(Q
ds as the separation of
Q from P; since ds is always positive, s will increase continuously
as Q moves along the world-line without retracing its steps,
so that Riemann's method is adequate. But suppose, that .hav-
ing laid down such a set of quantities g^ and thus associated
a value of ds with each interval, we arbitrarily and non-
uniformly stretch our measure of the separation of the events
along a certain curve. Suppose that, in mathematical lan-
guage, fixing attention on a certain defined world-line we write
da = <j>ds
where (f> is an arbitrary positive multiplier varying in any
manner from point to point along the world-line. Then Ida
varies always in the same sense as Ids ; and therefore for the
single purpose of describing the order of events on this world-
line I dcr is as adequate as ids.
FURTHER GENERALIZATION (WEYVS THEORY} 133
Also as long as we are thinking only of one particular
world-line we are concerned only with the values of </> along
that line. We are not bound in considering two world-lines
which intersect one another to limit ourselves to the same
quantity <£ at the common event. For defmiteness all that is
necessary is that each world-line shall itself determine what
value of (/> is to be taken at each point of itself.
Now just as Riemann found a sufficient means of prescrib-
ing a measure for the interval between adjacent events by
means of the quantities g^v and the quadratic form, we are
able to devise a sufficient means for our new purpose by lay-
ing down the rate at which (/> varies in any direction starting
from an arbitrary event. This is done by putting
S(log<£) = 2<^^(/4= 1,2, 3,4)
where ^ is a set of four arbitrary functions.
If further we allow that when the co-ordinates are changed
(^ is also to be so changed that the linear form $^dx^ remains
invariant, then the values of </> to which it leads along any
world-line are not affected by any change of co-ordinates.
105. EINSTEIN'S INTERVAL BETWEEN Two EVENTS NO
LONGER A DEFINITE QUANTITY.
Suppose then that this linear form is laid down in an
arbitrary manner. In general, starting from a given value <f>0
at P, the value of </> at another point Q will depend upon the
path by which Q is reached. This will be so unless Uy£rM
taken round any closed curve is zero1 — that is unless
~^ = ~ at all points. But the purpose we have in view
O Kv U^fj.
does not need that this should be so, inasmuch as we are
only concerned with the order of points along one world-line
at a time.
The gradients <£M rank with the coefficients g^v. We may
lay down their values arbitrarily for the whole of the field.
/Q TQ
QiJidXfj.. If I <f>fj.dXfj. has the same value for two paths
say PAQ, and PBQ; then I fadx^ for the path PAQBP is zero; and vice versa.
134 RELATIVITY AND THE ELECTRON THEORY
But when this is done, the multiplier 0 is not as a rule de-
fined except as to its variation along any given world-line.
Thus the interval dcr = <$)ds is not a unique and defined quantity .
Thus with this generalization the argument for a shift in the
spectral lines due to a gravitational field does not hold
(see § 95).
1 06. REDUNDANCE IN THE ARBITRARINESS OF THE
MODE OF ORDERING.
There is a certain amount of overlapping however in the
arbitrariness of the g^ field and the <£ field.
We do not alter the value of the separation do- for every
element of any given curve if we multiply the value of (j> by
an arbitrary function of position -\/r, and at the same time
divide each of the quantities g^ by i|r2.
Thus the ordering of events by this method is one in
which the measure relations remain unaltered if we put
Further we do not change the value of each element of separa-
tion if we replace the co-ordinates x^ by any other set of co-
ordinates x^ provided that at the same time the coefficients
fa and g^v are replaced by suitable other quantities $^ and g^
such that the two differential forms fa dx^ and g^v dx^ dx^,
are both invariant. In the language of the ordinary tensor
theory employed by Einstein, this means that fa is to be a
covariant vector and g^v the ordinary covariant tensor of second
rank.
We may say that <p^ and g^ define a measure -system, but
a given measure-system leaves <£M and g^ largely arbitrary.
1 Putting </>' = $<(>,
we have
=-~r (log
FURTHER GENERALIZATION (WEYUS THEORY] 135
107. NATURE DOES NOT MAKE QUANTITATIVE
EVALUATIONS.
Thus much for the mere machinery of recording the order
of events. Next as to the nature of the relations which de-
scribe natural phenomena.
The observed relations, as has been said, are merely of
coincidence. But they are always mathematically recorded
by means of numerical quantities introduced for the purpose.
Nature itself cannot recognize or prescribe any particular method
of quantitative evaluation ; that is only part of our intellectual
machinery.
If, therefore, we write down any mathematical equation
which expresses observed relations it must be of such a nature
that its validity is independent of the arbitrary choices of the
quantities used for the purpose of mathematical expression.
It should be true, therefore, that any order relations will
be maintained true if we make any arbitrary change in the
method by which we assign to any pair of events a numerical
measure of the interval between them.
1 08. INTRINSIC QUALITIES OF A MEASURE-SYSTEM.
DIRECT ROUTES. INVARIANTS.
If we adopt an arbitrary measure-system, that is, if we
assign certain numerical magnitudes to the intervals of each
world-line, as in §§ 104-6 we have a complete relativity of co-
ordinates within such a measure-system, as well as an arbi-
trariness in what we may call the guage-system <^.
But each measure-system has intrinsic characteristics indepen-
dent of the choice of co-ordinate-system or gauge-system. As an
illustration of this, suppose that we have assigned an arbitrary
field of values to gj and ^M, and adopted an arbitrary system
of co-ordinates for the mapping out of our four-dimensional
world. Let A, B be any two events, and </>0 the value of <£ at
x\. Then for any given path between A and B, I dcr is defined.
It follows that the direct routes from event to event are
136 RELATIVITY AND THE ELECTRON THEORY
thereby laid down, a direct route being defined as one for
which \d(j has a stationary value. Thus each measure- system
has its characteristic system of direct routes or geodesies, the
determination of which depends only upon the two invariants
fa dXfi and g^ dx^ dxv.
Each measure-system may also have associated with it
certain quantities whose values are unaltered under co-ordinate
changes and guage-changes which leave any given interval
unaltered. For instance the integral
where /« - - , and /* - ^ ^ /„
taken over any given region in the four-dimensional world is
such an invariant. Another quantity so invariant is I </>M dx^
taken round any closed curve ; for, on the understanding that
^ is a covariant vector <j>^ dx^ is invariant for co-ordinate
changes, and for a change of guage as above (§ 106), we have
f </,,
since the total increment in log i/r is zero on describing a
closed circuit.
109. PHYSICAL LAWS AND SPECIAL MEASURE-SYSTEMS
Einstein's theory and its extension at the hands of Weyl
seek now to establish a correspondence between these intrinsic
properties of the measure-system and the universe of material
phenomena.
For instance, part of Einstein's hypothesis for the explan-
ation of planetary motion is that the free motion of a particle
is a direct route or geodesic of the measure-system. We may put
this hypothesis in the form of a question — can we, out of all
possible measure-systems, select one in which the direct routes are
the actual paths of free particles ?
Further, Einstein seeks to set up a correspondence between
FURTHER GENERALIZATION (WE YDS THEORY} 137
the presence of matter, the gravitational influence of matter
and the intrinsic properties of the measure-system ; with this
in view he selects the tensor, G^v, propounding the hypothesis
that the vanishing of this is to correspond with the absence
of matter.
Thus we have another question ; is it possible to select from
all possible measure-systems one which is such that the tensor, G^v,
vanishes at all regions in the four-dimensional world which are
not occupied by matter ?
To both of these questions the provisional answer is " yes " ;
and the experiments have confirmed the answer by showing
that its consequences in the region, open to test, are actually
observed. Thus the law of gravitation becomes the condition
which singles out a measure-system. We have seen in the last
chapter how the law G^u = o, with the assumption of symmetry
round a centre, did actually lead to a definite measure-system
in which the path of Mercury is verified in practice to be a
geodesic.
no. NATURAL MEASURE-SYSTEMS.
What does this mean for other measure-systems in
which the path of Mercury is not a direct route? Mathe-
matically they are equally capable of describing the relative
positions and orders of the four-dimensional world. But in
the picture which they would give of it, there would be a
confusion between the natural geodesies or freepaths of par-
ticles, and the direct routes of the measure-system ; between
the regions occupied by matter and the regions of curvature
of the measure-system. The confusion would be of the same
type as that which in the old Newtonian theory would exist
between actual forces producing accelerations and apparent
forces arising if the motion of the frame of reference was
unknown. Thus it appears that among all possible measure-
systems there are some in which there is a specially simple
form for the order relations inherent in the physical world.
These might be called natural measure-systems.
Prof. Eddington states the conclusion with regard to
natural measure-systems differently. What has been called
138 RELATIVITY AND THE ELECTRON THEORY
here a measure-system, he calls a " species of space-time ". He
asks if every kind of space-time can occur in nature. To this
he replies, " No ! Einstein's law is a law which determines those
kinds of space-time which can occur in nature". This sug-
gests that material phenomena are incapable of description
on the background of any kind of space-time not satisfying
that law. But if the view expressed above is correct, this is
not so.
There is of course no such thing as a measured space-
time. We may speak of measuring space in the ordinary ex-
perimental sense ; but the process of measuring is itself a
sorting out of certain relations in the complex of phenomena.
In the four-dimensional world in which all phenomena are
laid out as a whole, any experiment of measuring space is
laid out as part of the picture, and the observations are simply
of the fact that certain world lines intersect (see § 99).
We must interpret Prof. Eddington's assertion that " only
certain kinds of space-time occur in nature" in the sense
that only certain kinds show that close correspondence
with nature which enables us to identify matter with aspects
of curvature. But we must remember that, in making such
an identification, we are simply resuming in picturesque
language the fact that experiment confirms Einstein's law.
Curvature of a four-dimensional manifold is a mere arith-
metical conception, it is form without content. The process
of identification is simply a recognition that the order
relations of the physical world are of that mathematical type
which we find in the algebraic theory which, by analogy with
the early ideas of curvature, we call the theory of curvature
in four-dimension or five-dimension space.
in. LEAST ACTION.
It has been remarked in chapter vii., p. no, that Einstein's
law of gravitation follows from the principle of least action, if
we take as the action the expression I G Jg dx^ dx^ dxz dx^
G being the scalar invariant of curvature, which, we may re-
mind ourselves, is a function of g^v and their first and second
FURTHER GENERALIZATION (WEYDS THEORY} 139
differential coefficients. In applying the principle of least
action the method is to calculate the variation of this expres-
sion for arbitrary changes in g^v.
Writing SA = | G^Sg*", it is inferred that if SA = o
for an arbitrary small change in the tensor g*v, then we must
have GMV = o. This stands as an equation, limiting the arbit-
rariness otherwise allowable in the tensor £• .
The principle of least action in its physical aspect would
then say that the happenings in nature are such as, when
interpreted in terms of a gravitational field g^n cause that field
to be such that this function of its whole condition, the action
has a least (or rather stationary) value.
On the geometrical side, however, if we think of g^v with a
set of co-ordinates as constituting a measure-system, the law
simply stands as a principle of selection among the measure-
systems. We arrive at the conclusion that, in order to obtain
the simplest and most complete correspondence between geometry
and natural law, we have to choose our measure-system in such
a way that the total curvature, I G *]g dx has a least value.
Here of course we are not considering the further general-
ization of this chapter.
112. WEYL'S THEORY OF ELECTRICITY.
In Einstein's theory of gravitation as described in the last
chapter, a certain expression GMI, depending on g^v and its
differential coefficients was described. The vanishing of GMI,
at a point, indicated the absence of matter at that point. The
coefficients g^v may be taken as characteristic of the gravi-
tational field, and GM|/ as measuring the nature of the matter
at any point.
We have now contemplated four other quantities ^ In
their origin, like g^ they arose out of the attempt to
express order relations in a quantitative form. But the
possibility of establishing a close correspondence between the
properties of matter and those of the measure-system enabled
us to give to g^v a physical significance. In the same way any
140 RELATIVITY AND THE ELECTRON THEORY
correspondence that may be set up and experimentally con-
firmed between some property of the measure system involving
<£M and some property of matter will enable us to give a
physical significance to 0M.
Now it was mentioned above that I fy^dx^ taken round any
closed curve is a quantity fixed within a given measure-
system. There may for instance be regions which are such
that this quantity vanishes for any closed curve drawn within
the region. This will be a property of such a region whatever
changes are made in the co-ordinate system or in <^, subject
only to the measures of the intervals being unvaried.
But if the g^ and ^ are varied in an arbitrary manner,
clearly \^>^dx^ will not be an invariant.
Now in Weyl's theory any region which is such that
vanishes for all curves drawn within it corresponds to
\
one which is free from electric and magnetic force. Effectively
this means that the four quantities are identified with the
electro-magnetic potentials,1 or conversely that in a field which
is free from electric and magnetic force </> has a unique value at
end pointy and therefore the interval do- has a definite value.
Here as in the gravitational case we are recognizing simply
a community of form, which makes it possible for us to
identify intrinsic qualities of a measure-system with qualities
of the physical universe. Their values will be relative to the
co-ordinate system adopted.
1 Thus : if we put/Vfe = — - *» --^p-» the necessary and sufficient conditions
9*fc d*»
that I 0/i dXfi. = o, for any curve within a region are that/i& = o for all values of
i and k, at all points of the region.
Putting fa = (F, G, H, and *) we have
c>F
43
which enable us to identify F, G, H, with the vector potential and * as the
scalar potential, (/14, fw /34) being the electric intensity and (/28i/si»/ia) *he
magnetic intensity,
FURTHER GENERALIZATION (WEYLS THEORY] 141
Further, in the regions in which l^/^ round a closed
curve does not vanish, it is possible out of the functions </>M to
construct a farther set of quantities SM, the relation of which
to <p^ is identical with the relation of the stream vector p (ux,
uy, ux, c) to the electro-magnetic potentials (see p. 84). The
relation between S^ and $M is preserved in any arbitrary
change of the measure system. The tensor S^ represents an
intrinsic quality of the measure-system.
Weyl's hypothesis is that it is possible so to choose the
measure system that the regions in which S^ = o correspond
exactly to the regions in which there is no electricity. This is
the complete analogue of Einstein's hypothesis that GM|/ = o
corresponds to the absence of matter.
The justification for Weyl's hypothesis is simply the
community of form between the relations SM = o and Max-
well's equations. They are indeed identical. So far there is
no experimental test proposed of the accuracy of the hypo-
thesis. But there is much to recommend it. The main
points in its favour are these : —
(i) The analysis proposed is forced upon us by the
incompleteness of Riemann's point of view.
(ii) The attempt to carry out the generalization makes
a natural place for a tensor ^ corresponding to the
electro-magnetic potentials in addition to the tensor
gpr which we have seen can represent the gravi-
tational field.
Thus electricity and gravitation are placed side by side as
fundamental properties of our perceived universe, common to
all kinds of matter.
It will to some extent be a matter of the predilection of
the student of these matters, whether he will now consider
that the physical laws of the universe are reduced to geomet-
rical laws or the reverse. It is clear that the course of nature
is more closely related to some systems of measurement than
to others — just as it was when all our exact laws were
comprised in Newtonian dynamics. On the other hand, the
fact that we are able to make such an excellent guess at the
142 RELATIVITY AND THE ELECTRON THEORY
form of the laws of nature by mere geometrical considerations,
leaves us with an uneasy feeling that we are looking at nature
through coloured glasses, and are simply seeing those aspects
of phenomena which are the easiest for us to appreciate.
This however is the usual course of science, and it is something
to feel that this most comprehensive of physical theories itself
suggests that when our power of vision is greater we shall
discover whole new realms of phenomena.
113. THE CONSERVATION OF ELECTRIC CHARGE.
It has been said that in Weyl's theory a certain tensor
SM is identified with the stream vector of electricity. Now we
find that when the variables <£M, g^ and the co-ordinates are
altered in any manner consistent with leaving the interval
measures unchanged, the tensor S^ satisfies an identical invari-
ant relation. This is completely analogous to the four identi-
ties connecting the quantities GMV when the element grs dxv dxs
is left invariant (cf. § 89). These four relations, it was said
above, are the relations which embody the principles of con-
servation of energy and momentum, energy and momentum
being special aspects of the tensor GMl/ which emerge when
we resolve our four-dimensional world into the special
aspects space and time. So here, when SM is interpreted in
the space-time aspects of electric charge in motion, the
identical relation which it satisfies as a consequence of relativity
stands as expressing the principle of the conservation of electric
charge.
114. WEYL'S THEORY AND THE ATOMIC NATURE OF
ELECTRICITY.
At this point we become conscious of the limits of Weyl's
theory as so far expounded.
The hypothesis is that it is possible to choose certain
measure-systems so that certain intrinsic properties of the
system correspond to the world lines of the electric charges.
FURTHER GENERALIZATION (WE YL'S THEORY] 143
In the light of the electron theory, the function of the equations
of the electric field is to co-ordinate the motions of electrons,
to predict the motion of an electron in a given field. If we
think of electricity as distributed in nuclei of extremely small
dimensions, we shall have the equations
SM-o
as characteristic of the field external to the electrons. These
being identical in form with Maxwell's equations will lead to
the usual expressions for the field due to electrons having
prescribed motions.
But it has already been pointed out (p. 24) that those
equations are insufficient to determine the motion of any
electron without some further information as to the structure
of the electron (e.g. Lorentz' contracting electron) or some
equivalent hypothesis. We have seen that the Special
Principle of Relativity was able to fill the gap for the slowly
accelerated electron.
If we seek to supplement Weyl's equation by some
further hypothesis, we have this elementary treatment to
guide us, but we have also the method adopted by Einstein
with such success for the motion of the material particle.
We recall that having by means of the equations GMV = o
determined the field of a certain specified distribution of
matter, the path of the free particle was taken to be a geodesic
of this field.
This would be a possible method of attack in the electrical
case. Having obtained the electric field of a given distribution
of matter and electricity (we cannot now separate the two) by
means of the equations SM = o, we could make the provisional
hypothesis that in this field the world line of a free electron
is a direct route, that is a path for which \dcr is stationary.
This would make the problem of determining its motion a
determinate one, would involve no special analysis of the
constitution of the electron, and it would be consistent with
the results of the special principle which have been verified
experimentally. But this is speculation, and the suggestion
has not yet been explored.
144 RELATIVITY AND THE ELECTRON THEORY
115. SUMMARY OF MATHEMATICAL SPECIFICATION OF
WEYL'S THEORY.
,„ 30* 30*
fik = o — ^z— .
C#* C*t
/** = ^ g**M.
f»
Definition of stream of electricity
'
Compare with this Minkowski's equations, p. 84.
These give |?*= o
which is the equation of conservation of electricity.
If there is no gravitation grs= o if r=£s, also g-n = gZ2 = ^33 = - i, and
£44 = + *• Tne equations then reduce exactly to Maxwell equations.
The expression for the action in a region not occupied by charge is
flf* f« dr
and this integral is proved to be an absolute invariant for any change in co-
ordinates, or in the potentials </>M, subject to the intervals being unvaried.
1 1 6. COMPARATIVE SURVEY.
It may be worth while to set down here a comparison of
the conclusions from Newton's theory of dynamics, Einstein's
restricted Principle of Relativity, Einstein's generalized theory,
and finally of Weyl's theory.
I. NEWTON. — In the region of dynamics, of all possible
co-ordinate systems there is a limited group for which the
laws of motion have a particularly simple form. In any
system of the group the path of a free particle is a straight
line described uniformly.
II. EINSTEIN, 1905 — Restricted Principle of Relativity. —
In the region of dynamics and electro-dynamics, of all possible
co-ordinate systems there is a limited group for which the laws
are of precisely the same form. In any system of this group
the path of a free particle is a straight line described uni-
formly, and light travels with constant velocity.
III. EINSTEIN, 1915 — General Principle of Relativity. — In
the region of gravitational phenomena, there is no restriction
on the co-ordinate system, but the measure-system is limited
by the hypothesis (i) that the world-line of a free particle is a
direct line of the measure-system, (ii) that the curvature GMI,
FURTHER GENERALIZATION (WEYUS THEORY] 145
may be identified with the presence of matter. The interval
between two events is a fixed quantity within the appropriate
measure-system.
IV. WEYL's THEORY. — In electro-dynamic as in gravi-
tational phenomena, there is no restriction on the co-ordinate
system, and the measure of the interval between two events
is not a definite quantity determined by the gravitation field.
But of all possible measure-systems, there is one, or a limited
group, in which the singular regions of the system are in exact
correspondence with the regions in which matter or electricity
are present.
We conclude then that Einstein has satisfied the demand
that nature itself shall not show any preference for any
particular system of variables or co-ordinates by which we
shall distinguish between events. Further, Weyl has made it
clear that the laws of nature do not supply us with an absolute
criterion of equality of intervals of space and time, except
that of coincidence ; in fact the idea of a definite measurable
interval between two events has to some extent broken down.
But we are not left with the conclusion that we may adopt an
absolutely arbitrary system of measurement. There is neces-
sarily one measure-system or a limited group, within which
there is complete relativity of co-ordinates, in which there is a
particularly simple correspondence between the geometry of
the system and nature ; or in other words, for which the
mathematical relations of nature take the simplest possible
form.
In the end we must admit therefore that the recognition
of order in the sequences of events around us arises from an
adjustment of mental machinery to the events. Just as we
only recognize a star as a sharp point of light when the eye
focusses the light on the retina, so we recognize a distinct
order in the universe when we focus our measuring system
properly. The eye has an infinity of ways of focussing itself
so that the star produces a blurred image on the retina. So
the mind may form many images of nature which are in a
sense blurred, in which a distinct order is not perceptible.
But the fact that the eye can produce a sharp image is enough
146 RELATIVITY AND THE ELECTRON THEORY
both to determine our conception of the star, and to define
what we mean by the proper focus for the star. So the
problem of science is mainly to discover that mental focus in
which nature gives us clear impressions of order. Einstein
has shown that this may be done in a way far more compre-
hensive than had been thought possible. He has shown us
that the required mental focus is to be attained without any
introduction of metaphysical notions such as those of an
absolute space and time, or even of such a remnant of those
notions as that light shall have a definite velocity the same at
all points.
INDEX.
ABERRATION, 16.
Abraham, 38, 70, 81.
Absolute, time, space, velocity, 8.
Action, principle of, 77, 83, 108, no,
138.
Addition of velocities, Einstein, 39.
Newton, n.
yEther, concealment cf, 48.
— mechanics of, 89.
— momentum in, 81.
— penetrates matter, Fresnel, 16.
Arago, 13.
Atomic nature of electricity, 52, 142.
BESTELMEYER, A., 66.
/3-rays, 64.
Brace, D. B., see Rayleigh, 56.
Bucherer, A., 66.
CATHODE-RAYS, 38, 68.
Christoffel, 100.
Clifford, W. K., 100.
Coincidence, in experimental observa-
tion, 29, 50, 93 126.
Conservation, oi electricity, 48, 142.
— of energy and momentum, 12, 79,
in.
— of mass, 48.
Contracting electron, 25, 56, 70.
Contraction hypothesis, 23, 23, 44.
— Larmor, 24.
— Lorentz, 24, 55.
Convection coefficient, 41, 43.
Co-ordinate systems, and propagation
of light, 95.
relativity of, 93.
Couple on moving condenser, 59.
Curl of a vector, i.
Curvature, in four dimensions, 103.
— invariant of, 108-9.
— specific, 102.
DEFLECTION of light, 119.
De Sitter, 86.
Direct routes, 135.
Dispersive media, convection coefficient
in, 43.
Doppler effect, 42.
Dynamics, Newtonian, 8, 10, 39, 78, 83.
ECLIPSE Expedition, 121, 127, 131.
Economy of thought, 90.
Eddington, 50, 112, 121, 137.
Electric intensity, relativity of, 46.
Electric intensity, transformation of, 48.
Electricity, 46.
— atomic nature of, 52, 142.
— conservation of, 48, 142.
— Weyl's theory of, 139.
Electro-magnetic field, equations, 46,
84.
— inertia, 25, 70.
Electron, apparent mass of, 68, 70.
— finite size of, 53.
— path of, 143.
— theory of matter, 24, 52.
Energy, and mass, 78.
— and momentum, 79.
— conservation of, in.
Eotvos, 87.
Euclidean geometry, 102.
FITZGERALD, contraction hypothesis,
22, 31, 57.
Fizeau experiment, 14, 16, 41.
Force, and work, 78.
generalized, 78.
— mechanical, 45.
— on moving charged bodies, 59.
— relativity of, 96.
Four-dimensional geometry, 103.
Fresnel, and Arago, 13.
— convection coefficient, 14, 41.
GALILEO, 78.
Gauss, 109.
Geodesic, 107, 143.
Geometry, bases of, Riemann, 100.
— Euclidean, 102.
— four-dimensional, 103.
— non-Euc idean, 123.
Gravitation, and electrical theory of
matter, 27, 86.
— and light, 89.
— as a means of communication, 38.
— Einstein's Law, 104, 105.
— real and simulated, 97.
Gravitational, field round the sun, 113.
— mass, 87.
Guage-system, 135.
HAMILTON'S principle, 108, no.
Hertz, 17.
Hupka, 68, 70.
Huyghens, 78.
Hypothesis of constant light-velocity,
28.
147
148 RELATIVITY AND THE ELECTRON THEORY
iNERTiA-mass, 87.
Interval between events, 121, 133.
Invariant, action, 83, 112.
— charge, 12.
— element of integration, 109.
— of curvature, 108, 139.
KAUFMANN'S experiments, 25, 64.
LARMOR, contraction hypothesis, 24.
— point electrons, 53.
— Rayleigh-Brace experiment, 57.
— space-time transformations, 35.
Length, dependent on velocity, 31.
Levi-Civita, too.
Light, constant velocity, 33.
— deflection of, 119.
— gravitational effect on, 89.
MAGNETIC intensity, relativity of, 46.
Maps, 91.
Mass and energy, 80.
— apparent, of electrons, 67.
Matter, gravitational field in absence of,
105.
— modified by motion through the
aether, 14, 17.
— tensor, in.
Maxwell, 17, 22, 87.
Maxwell's equations in Weyl's theory,
141.
Measure-systems, 134.
— correspondence with nature, 136.
Measuring-rod, 127.
Mechanical theory derived from electri-
cal, 61.
Mechanics, 78.
Mercury, motion of perihelion, 113,117.
Michelson, A. A., repetition of Fizeau's
experiment, 6, 42, 44.
Michelson-Morley experiment, 17, 19,
21, 23, 37, 54.
Minkowski, 72, 83, 84, 94.
Momentum and energy, 79.
— conservation of, 12, in.
— of aether, 81.
Morley, E. W., see Michelson, A. A.
— and Miller, D. B., 18, 22, 23.
Motion, analysis of concept, 10.
— of charged particle, 62.
— of free particle and propagation of
light, 96.
— of material particle, 80.
NEUMANN-Schaefer, 68, 71.
Newtonian dynamics, 8, 10, 38, 129, 144,
Newtonian potential, 99.
Noble, see Trouton, 39, 58.
Non-Euclidean geometry, 123.
Nordstrom, 86.
OBJECTIONS to the Principle of Relativ^
ity, 38. 4i.
Observations, nature of, 93, 126.
PATH of free particle has stationary
length, 96, 106.
Perihelion of Mercury, 113.
Planet, law of motion of, 115.
Poincare, 86.
Potential, Newtonian, 99.
Poynting, flux of energy, 81.
Propagation of light, 95-6.
QUADRATIC Form, 94.
RANKINE, A. O., see Trouton, F. T.,
Rayleigh, Lord, and Brace, D. B., 56.
on double refraction in mov-
ing bodies, 25.
Relative velocity, 40.
Riemann, 100, 132.
Ricci, 100.
Rigid body, 130.
ScHAEFER-Neumann, 68, 71.
Simultaneity, 29.
Soddy, 41.
Space and time, absolute, 8.
unified, 75.
Space-time transformations, Lorentz,
34, 73-
Newtonian, 10.
Spectral-shift, suggested, 121.
Stokes, Sir G. G., on aberration, 16.
Sun, gravitational field of, 113.
TENSOR, defining kind or space-time,
97-
— of matter or energy, in.
Thomson, J. J., 70.
Transformations, Lorentz, 34, 73.
— Newton, 10.
— of charge, 48.
— of electrical intensity, 47.
Trouton, and Noble, 39, 58, 82.
— and Rankine, 57.
VECTORS, definitions and notation, i.
— in four dimensions, 72.
Velocity, absolute, 8.
— addition of, Einstein, 39.
Newton, n.
— of light, constant, 33.
critical, 40.
in moving matter, 13.
Velocity, relative, 40.
Verification of Einstein's theory, 112.
WEYL, 125, 139, 142, 145.
Wolz, 66.
World-line, length of, 106.
Wright, J. E., 100.
ZEEMAN, and Fizeau's experiment, 44.
— effect, 68.
— Eotvos' experiment, 88.
RETURN CHEMISTRY LIBRARY
TO— •* 1 00 Hildebrand Hall
642-3753
LOAN PERIOD 1
21 MONTH
ALL BOOKS MAY BE RECALLED AFTER 7 DAYS
Renewable by telephone
DUE AS STAMPED BELOW
UNIVERSITY OF CALIFORNIA, BERKELEY
FORM NO. DD5, 3m, 12/80 BERKELEY, CA 94720
GK 6
u <
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11